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THE  CONTINUUM 

AND  OTHER  TYPES  OF  SERIAL  ORDER 


WITH  AN  INTRODUCTION  TO  CANTOR'S 
TRANSFINITE  NUMBERS 


BY 
EDWARD  V.  HUNTINGTON 

ASSOCIATE  PROFESSOR  OF  MATHEMATICS 
IN  HARVARD  UNIVERSITY 


SECOND  EDITION 


HARVARD  UNIVERSITY  PRESS 

CAMBRIDGE,  MASS.,  U.S.A. 
1917 


COPTKIGHT,  1917 
HABVABD  UNIVEBSITY  PRESS 


PREFACE  TO  THE  SECOND  EDITION 

The  first  edition  of  this  book  appeared  in  1905  as  a  reprint 
from  the  Annals  of  Mathematics,  series  2  (vol.  6,  pp.  151-184, 
and  vol.  7,  pp.  15-43),  under  the  title:  The  Continuum  as  a 
Type  of  Order:  an  Exposition  of  the  Modern  Theory;  with  an 
Appendix  on  the  Transfinite  Numbers  (The  Publication  Office 
of  Harvard  University,  Cambridge,  Mass.). 

An  Esperanto  translation  by  R.  Bricard,  under  the  title: 
La  Kontinuo,  appeared  in  1907  (Paris,  Gauthier-Villars). 

The  following  reviews  (of  the  original  or  of  the  translation) 
may  be  noted:  by  O.  Veblen,  in  Bull.  Amer.  Math.  Soc.,  vol. 
12  (1906),  pp.  302-305;  by  P.  E.  B.  Jourdain,  in  the  Mathe- 
matical Gazette,  vol.  3  (1906),  pp.  348-349;  by  C.  Bourlet,  in 
Nouvelles  Annates  de  Mathematiques,  ser.  4,  vol.  7  (1907), 
pp.  174-176;  and  by  Hans  Hahn,  in  Monatsheftefur  Math.  u. 
Physik,  vol.  21  (1910),  Literaturher.,  p.  26.  The  author  is 
indebted  to  Professor  Veblen  and  to  Professor  Hahn  for 
calling  his  attention  to  errors  in  §  62. 

The  principal  modifications  in  the  present  edition  are  the 
following :  §  38  and  §  64  have  been  enlarged ;  §  62  has  been 
rewritten,  and  §  62a  has  been  added;  the  bibhographical 
notes  have  been  brought  more  nearly  up  to  date;  through- 
out Chapter  VII  [formerly  called  the  Appendix  (§  73-§  91)] 
the  term  ''normal  series"  has  been  replaced  by  the  term 
"  well-ordered  series  "  (for  reasons  explained  in  a  footnote 
to  §  74) ;  and  in  §  89a  a  brief  account  has  been  inserted  of 
Hartogs's  recent  proof  of  Zermelo's  theorem  that  every  class 
can  be  well-ordered. 


CONTENTS 


Introduction 


CHAPTER  I 

On  Classes  in  general 

SECTION 

1-6.    One-to-one  correspondence  between  two  classes 4 

7-10.     Finite  and  infinite  classes 6 

11.    The  simplest  mathematical  systems 8 

CHAPTER  II 

On  simply  ordered  Classes,  or  Series 

12-15.  Definition  of  a  series:  Postulates  1-3 10 

16.  Ordinal  correspondence  between  two  series 11 

17-18.  Properties  of  series.     Successor.     Predecessor    ......  12 

19.  Examples  of  series 13 

20.  Examples  of  systems  which  are  not  series 16 

CHAPTER  III 

Discrete  Series:  Especially  the  Type  u  of  the 

Natural  Numbers 

21-22.  Definition  of  a  discrete  series:  Postulates  Nl-NS 19 

23.  Theorem  of  mathematical  induction 20 

24-27.  Properties  of  discrete  series.   Progressions,  or  series  of  type  w  21 

28.  Examples  of  discrete  series;  the  natural  numbers 23 

29.  Examples  of  series  which  are  not  discrete 23 

30-36.  On  numbering  the  elements  of  a  discrete  series.    Sums  and 

products  of  the  elements     25 

37-40.    On  denumerable  classes 30 


vi 


CONTENTS 


CHAPTER  IV 

Dense  Series:  Especially  the  Type  rj  op  the 
Rational  Nltmbers 

41-43.  Definition  of  a  denumerable  dense  series :  Postulates //1-H2  34 

44-45.  Properties  of  denumerable  dense  series.     Series  of  type  77.  35 

46-50.  Segments.    Limits 37 

51.  Examples  of  denumerable  dense  series;  the  rational  numbers  39 

52.  Examples  of  series  which  are  not  dense 41 

53.  On  the  arithmetical  operations  among  the  elements  of  a  dense 

series 43 

CHAPTER  V 

Continuous  Series:  Especially  the  Type  6  of  the 
Real  Numbers 

54-55.  Definition  of  a  linear  continuous  series:  Postulates  C1-C3  .  44 

56-62.  Properties  of  linear  continuous  series.     Series  of  type  6  .    .  45 

62a.  Digression  on  the  theory  of  sets  of  points 52 

63.  Examples  of  linear  continuous  series;  the  real  numbers    .    .  52 

64.  Examples  of  series  which  are  not  continuous 55 

65.  On  the  arithmetical  operations  among  the  elements  of  a  con- 

tinuous series 57 


CHAPTER  VI 

Continuous  Series  of  More  than  One  Dimension,  with 
a  Note  on  Multiply  Ordered  Classes 

66-70.     Definition  of  n-dimensional   continuous  series.     Series  of 

type0" 58 

71.  The  one-to-one  correspondence  between  the  points  of  all 

space  and  the  points  of  a  line 60 

72.  Note  on  multiply  ordered  classes 62 


CONTENTS  vii 


CHAPTER  VII 


Well-ordered  Series,  with  an  Introduction  to 
Cantor's  Transfinite  Numbers 

73-76.    Definition  of  a  well-ordered  series :  Postulates  4-6    ....  63 

77-85.    Examples  and  properties  of  well-ordered  series 66 

86.    The  transfinite  ordinal  numbers;   u,  U,  etc 73 

87-91.    The  transfinite  cardinal  numbers;    Xo,  Ni,  etc 74 

Index  of  technical  terms 81 


THE  CONTINUUM 

AND  OTHER  TYPES  OF  SERIAL  ORDER 


INTRODUCTION 

The  main  object  of  this  book  is  to  give  a  systematic  elementary 
account  of  the  modern  theory  of  the  continuum  as  a  type  of  serial 
order  —  a  theory  which  underhes  the  definition  of  irrational  num- 
bers and  makes  possible  a  rigorous  treatment  of  the  real  number 
system  of  algebra. 

The  mathematical  theory  of  the  continuous  independent  vari- 
able, in  anything  like  a  rigorous  form,  may  be  said  to  date  from  the 
year  1872,  when  Dedekind's  Stetigkeit  und  irraiionale  Zahlen  ap- 
peared;* and  it  reached  a  certain  completion  in  1895,  when  the  first 
part  of  Cantor's  Beitrdge  zur  Begriindung  der  transfiniten  Mengen- 
lehre  was  published  in  the  Mathematische  Annalen.'f 

While  all  earlier  discussions  of  continuity  had  been  based  more  or 
less  consciously  on  the  notions  of  distance,  number,  or  magnitude, 
the  Dedekind-Cantor  theory  is  based  solely  on  the  relation  of  order. 
The  fact  that  a  complete  definition  of  the  continuum  has  thus  been 
given  in  terms  of  order  alone  has  been  signaHzed  by  Russell  t  as  one 

*  Third  (unaltered)  edition,  1905;  English  translation  by  W.  W.  Beman, 
in  a  volume  called  Dedekind's  Essays  on  the  Theory  of  Numbers,  1901. 

t  Georg  Cantor,  Math.  Ann.,  vol.  46  (1895),  pp.  481-512;  French  translation 
by  F.  Marotte,  in  a  volume  called  Sur  les  fondements  de  la  theorie  des  ensembles 
trans  finis,  1899;  English  translation  by  P.  E.  B.  Jourdain,  Contributions  to  the 
Founding  of  the  Theory  of  Transfinite  Numbers,  Open  Court  Publishing  Co., 
1915.  For  further  references  to  Cantor's  work,  see  §  74.  An  interesting  con- 
tribution to  the  theory  has  been  made  by  O.  Veblen,  Definitions  in  terms  of 
order  alone  in  the  linear  continuum  and  in  well-ordered  sets,  Trans.  Amer.  Math. 
Soc,  vol.  6  (1905),  pp.  16^171. 

t  B.  Russell,  Principles  of  Mathematics,  vol.  1  (1903),  p.  303.  See  also  A. 
N.  Whitehead  and  B.  Russell,  Prindpia  Mathematica,  especially  vol.  2  (1912) 
and  vol.  3  (1913),  where  an  elaborate  account  of  the  theory  of  order  is  given 
in  the  eymbohc  notation  of  modem  mathematical  logic. 

1 


2  TYPES  OF  SERIAL  ORDER 

of  the  notable  achievements  of  modern  pure  mathematics ;  *  and  the 
simphcity  of  the  ordinal  theory,  which  requires  no  technical  knowl- 
edge of  mathematics  whatever,  renders  it  peculiarly  accessible  to 
the  increasing  number  of  non-mathematical  students  of  scientific 
method  who  wish  to  keep  in  touch  with  recent  developments  in  the 
logic  of  mathematics. 

The  present  work  has  therefore  been  prepared  with  the  needs  of 
such  students,  as  well  as  those  of  the  more  mathematical  reader,  in 
view;  the  mathematical  prerequisites  have  been  reduced  (except  in 
one  or  two  illustrative  examples)  to  a  knowledge  of  the  natural 
numbers,  1,  2,  3,  .  .  .  ,  and  the  simplest  facts  of  elementary  geom- 
etry; the  demonstrations  are  given  in  full,  the  longer  or  more 
difficult  ones  being  set  in  closer  type;  and  in  connection  with 
every  definition  numerous  examples  are  given,  to  illustrate,  in  a 
concrete  way,  not  only  the  systems  which  have,  but  also  those 
which  have  not,  the  property  in  question. 

Chapter  I  is  introductory,  concerned  chiefly  with  the  notion  of 
one-to-one  correspondence  between  two  classes  or  collections. 
Chapter  II  introduces  simply  ordered  classes,  or  series,t  and  ex- 
plains the  notion  of  an  ordinal  correspondence  between  two  series. 
Chapters  III  and  IV  concern  the  special  types  of  series  known  as 
discrete  and  dense,  and  chapter  V,  which  is  the  main  part  of  the 
book,  contains  the  definition  of  continuous  series.  Chapter  VI  is  a 
supplementary  chapter,  defining  multiply  ordered  classes,  and 
continuous  series  in  more  than  one  dimension.  Chapter  VII  gives 
a  brief  introduction  to  the  theory  of  the  so-called  "  well-ordered  " 
series,  and  Cantor's  transfinite  numbers.  An  index  of  all  the 
technical  terms  is  given  at  the  end  of  the  volume. 

*  The  fundamental  importance  of  the  subject  of  order  may  be  inferred 
from  the  fact  that  all  the  concepts  required  in  geometry  can  be  expressed  in 
terms  of  the  concept  of  order  alone;  see,  for  example,  O.  Veblen,  A  system 
of  axioms  for  geometry,  Trans.  Amer.  Math.  Soc,  vol.  5  (1904),  pp.  343- 
384;  or  E.  V.  Huntington,  A  set  of  postulates  for  abstract  geometry,  expressed  in 
terms  of  the  simple  relation  of  inclusion.  Math.  Ann.,  vol.  73  (1913),  pp.  522- 
559. 

t  The  word  series  is  here  used  not  in  the  technical  sense  of  a  sum  of  numeri- 
cal terms,  but  in  a  more  general  sense  explained  in  §  12. 


INTRODUCTION  3 

It  will  be  noticed  that  while  the  usual  treatment  of  the  con- 
tinuum in  mathematical  text-books  begins  with  a  discussion  of  the 
system  of  real  numbers,  the  present  theory  is  based  solely  on  a  set 
of  postulates  the  statement  of  which  is  entirely  independent  of 
numerical  concepts  (see  §  12,  §  21,  §  41,  and  §  54).  The  various 
number-systems  of  algebra  serve  merely  as  examples  of  systems 
which  satisfy  the  postulates  —  important  examples,  indeed,  but 
not  by  any  means  the  only  possible  ones,  as  may  be  seen  by  in- 
spection of  the  lists  of  examples  given  in  each  chapter  (§§  19,  28, 
51,  63).  For  the  benefit  of  the  non-mathematical  reader,  I  give  a 
detailed  explanation  of  each  of  the  number-systems  as  it  occurs,  in 
so  far  as  the  relation  of  order  is  concerned  (see  §  22  for  the  integers, 
§51, 3  for  the  rationals,  and  §63, 3  for  the  reals) ;  the  operations  of 
addition  and  multiphcation  are  mentioned  only  incidentally  (see 
§§  31,  53,  and  65),  since  they  are  not  relevant  to  the  purely  ordinal 
theory.* 

In  conclusion,  I  should  say  that  the  bibliographical  references 
throughout  the  book  are  not  intended  to  be  in  any  sense  exhaus- 
tive; for  the  most  part  they  serve  merely  to  indicate  the  sources  of 
my  own  information. 

*  The  reader  who  is  interested  in  these  extra-ordinal  aspects  of  algebra  may 
refer  to  my  paper  on  The  Fundamental  Laws  of  Addition  and  Multiplication 
in  Elementary  Algebra,  reprinted  from  the  Annals  of  Mathematics,  vol.  8  (1906), 
pp.  1-44  (  PubUcation  Office  of  Harvard  University) ;  or  to  my  Fundamental 
Propositions  of  Algebra,  being  monograph  IV  (pp.  149-207)  in  the  volume 
called  Monographs  on  Topics  of  Modem  Mathematics  relevant  to  the  Elementary 
Field,  edited  by  J.  W.  A.  Young  (Longmans,  Green  &  Co.,  1911).  A  more 
elementary  treatment  may  be  found  in  John  Wesley  Young's  Lectures  on 
Fundamental  Concepts  of  Algebra  and  Geometry  (MacmiUan,  1911). 


CHAPTER  I 

On  Classes  in  General 

1.  A  class  (Menge,  ensemble)  is  said  to  be  determined  by  any  test 
or  condition  which  every  entity  (in  the  universe  considered)  must 
either  satisfy  or  not  satisfy;  any  entity  which  satisfies  the  condi- 
tion is  said  to  belong  to  the  class,  and  is  called  an  element  of  the 
class.*  A  null  or  empty  class  corresponds  to  a  condition  which  is 
satisfied  by  no  entity  in  the  universe  considered. 

For  example,  the  class  of  prime  numbers  is  a  class  of  numbers 
determined  by  the  condition  that  every  number  which  belongs  to 
it  must  have  no  factors  other  than  itself  and  1.  Again,  the  class  of 
men  is  a  class  of  hving  beings  determined  by  certain  conditions  set 
forth  in  works  on  biology.  Finally,  the  class  of  perfect  square 
numbers  which  end  in  7  is  an  empty  class,  since  every  perfect  square 
number  must  end  in  0,  1,  4,  5,  6,  or  9. 

2.  If  two  elements  a  and  6  of  a  given  class  are  regarded  as  inter- 
changeable throughout  a  given  discussion,  they  are  said  to  be  equal; 
otherwise  they  are  said  to  be  distinct.  The  notations  commonly 
used  are  a  =  h  and  a  9^  b,  respectively. 

3.  A  one-to-one  correspondence  between  two  classes  is  said  to  be 
established  when  some  rule  is  given  whereby  each  element  of  one 
class  is  paired  with  one  and  only  one  element  of  the  other  class,  and 
reciprocally  each  element  of  the  second  class  is  paired  with  one  and 
only  one  element  of  the  first  class. 

For  example,  the  class  of  soldiers  in  an  army  can  be  put  into  one- 
to-one  correspondence  with  the  class  of  rifles  which  they  carry, 

•  H.  Weber,  Algebra,  vol.  1,  p.  4.  For  the  sake  of  uniformity  with  Peano's 
Formidaire  de  Mathematiques,  I  translate  Menge,  or  Mannigjaltigkeit,  by  class 
instead  of  by  collection,  mass,  set,  ensemble,  or  aggregate  —  all  of  which  terms 
are  in  use.  For  recent  discussions  of  the  concept  class,  see  the  articles  cited  in 
§83. 

4 


§4 


CLASSES  IN  GENERAL 


since  (as  we  suppose)  each  soldier  is  the  owner  of  one  and  only  one 
rifle,  and  each  rifle  is  the  property  of  one  and  only  one  soldier. 

Again,  the  class  of  natural  numbers  can  be  put  into  one-to-one 
correspondence  with  the  class  of  even  numbers,  since  each  natural 
number  is  half  of  some  particular  even  number  and  each  even 
number  is  double  some  particular  natural  number;  thus: 


1, 
2, 


2, 

4, 


3, 
6, 


Again,  the  class  of  points  on  a  line  AB  three  inches  long  can  be 
put  into  one-to-one  correspondence  with  the  class  of  points  on  a 


line  CD  one  inch  long;  for  example  by  means  of  projecting  rays 
drawn  from  a  point  0  as  in  the  figure. 

4.  An  example  of  a  relation  between  two  classes  which  is  not  a 
one-to-one  correspondence,  is  furnished  by  the  relation  of  owner- 
ship between  the  class  of  soldiers  and  the  class  of  shoes  which  they 
wear;  we  have  here  what  may  be  called  a  two-to-one  correspond- 
ence between  these  classes,  since  each  shoe  is  worn  by  one  and  only 
one  soldier,  while  each  soldier  wears  two  and  only  two  shoes.  The 
consideration  of  this  and  similar  examples  shows  that  all  the  con- 
ditions mentioned  in  the  definition  of  one-to-one  correspondence 
are  essential. 

*  That  the  class  of  square  niimbers  can  be  put  into  one-to-one  correspond- 
ence with  the  class  of  all  natural  numbers  was  known  to  Gahleo ;  see  his 
Dialogs  concerning  two  new  Sciences,  translation  by  Crew  and  de  Salvio  (1914), 
pp.  18-40. 


6 


TYPES  OF  SERIAL  ORDER 


§5 


5.  Obviously  if  two  classes  can  be  put  into  one-to-one  corre- 
spondence with  any  third  class,  they  can  be  put  into  one-to-one 
correspondence  with  each  other. 

6.  A  -part  ("  -proper  part,"  echter  Teil),  of  a  class  A  is  any  class 
which  contains  some  but  not  all  of  the  elements  of  A,  and  no  other 
element. 

A  subclass  (Teil)  of  A  is  any  class  every  element  of  which  belongs 
to  A ;  that  is,  a  subclass  is  either  a  part  or  the  whole. 

7.  We  now  come  to  the  definition  of  finite  and  infinite  classes. 

An  infinite  class  is  a  class  which  can  be  put  into  one-to-one  corre- 
spondence with  a  part  of  itself.  A  finite  class  is  then  defined  as  any 
class  which  is  not  infinite. 

This  fundamental  property  of  infinite  classes  was  clearly  stated 
in  B.  Bolzano's  Paradoxien  des  Unendlichen  (published  post- 
humously in  1850),  and  has  since  been  adopted  as  the  definition  of 
infinity  in  the  modern  theory  of  classes.* 

8.  An  example  of  an  infinite  class  is  the  class  of  the  natural 
numbers,  since  it  can  be  put  into  one-to-one  correspondence  with 
the  class  of  the  even  numbers,  which  is  only  a  part  of  itself  (§  3). 


Again,  the  class  of  points  on  a  line  AB  is  infinite,  since  it  can  be 
put  into  one-to-one  correspondence  with  the  class  of  points  on  a 
segment  CD  oi  AB  (by  first  putting  both  these  classes  into  one-to- 

*  See  G.  Cantor,  Crelle's  Journ.Jiir  Math.,  vol.  84  (1877),  p.  242;  and  espe- 
cially R.  Dedekind:  TFas  sind  und  was  sollen  die  Zahlen,  1887  (English  trans- 
lation by  W.  W.  Beman,  under  the  title  Essays  on  the  theory  of  Numbers,  1901) ; 


§  10  CLASSES  IN  GENERAL  7 

one  correspondence  with  the  class  of  points  on  an  auxiliary  line 
HK,  as  in  the  figure). 

The  class  of  the  first  n  natural  numbers,  on  the  other  hand,  is 
finite,  since  if  we  attempt  to  set  up  a  correspondence  between  the 
whole  class  and  any  one  of  its  parts,  we  shall  always  find  that  one 
or  more  elements  of  the  whole  class  will  be  left  over  after  all  the 
elements  of  the  partial  class  have  been  assigned  (see  §  27) . 

9.  The  most  important  elementary  theorems  in  regard  to  infinite 
classes  are  the  following: 

(1)  //  any  subclass  of  a  given  class  is  infinite  then  the  class  itself  is 
infinite. 

For,  let  A  be  the  given  class.  A'  the  infinite  subclass,  and  A"  the 
subclass  of  all  the  elements  of  A  which  do  not  belong  to  A'  (noting 
that  A"  may  be  a  null  class). 

By  hypothesis,  there  is  a  part,  A'l,  oi  A'  which  can  be  put  into 
one-to-one  correspondence  with  the  whole  of  A';  therefore  the  class 
composed  of  A'l  and  A"  will  be  a  part  of  A  which  can  be  put  into 
one-to-one  correspondence  with  the  whole  of  A. 

(2)  If  any  one  element  is  excluded  from  an  infinite  class,  the  remain- 
ing class  is  also  infinite. 

For,  let  A  be  the  given  class,  x  the  element  to  be  excluded,  and  B 
the  class  remaining.  By  hypothesis,  there  is  a  part,  Ai,oiA,  which 
can  be  put  into  one-to-one  correspondence  with  the  whole  of  A,  and 
is  therefore  itself  infinite.  If  this  part  Ai  does  not  contain  the  ele- 
ment X,  it  will  be  a  subclass  in  B,  and  our  theorem  is  proved.  If  it 
does  contain  x,  there  will  be  at  least  one  element  y  which  belongs  to 
B  and  not  toAi,  and  by  replacing  a;  by  y  in  A  i  we  shall  have  another 
part  of  A ,  say  A2,  which  will  be  an  infinite  part  of  A  and  at  the  same 
time  a  subclass  in  B. 

10.  As  a  corollary  of  this  last  theorem  we  see  that  no  infinite 
class  can  ever  be  exhausted  by  taking  away  its  elements  one  by  one. 

For,  the  class  which  remains  after  each  subtraction  is  always  an 
infinite  class,  by  §  9,  2,  and  therefore  can  never  be  an  empty  class, 

compare  B.  Russell,  Principles  of  Mathematics,  vol.  1,  p.  315,  and  Whitehead 
and  Russell,  Princijna  Mathematica,  vol.  2  (1912),  pp.  187-192.  See  also  §  27 
of  the  present  paper. 


8  TYPES  OF  SERIAL  ORDER  §  11 

or  a  class  containing  merely  a  single  element  (these  classes  being 
obviously  finite  according  to  the  definition  of  §  7). 

This  result  will  be  used  in  §  27,  below,  where  another,  more 
familiar,  definition  of  finite  and  infinite  classes  will  be  given. 

The  further  study  of  the  theory  of  classes  as  such,  leading  to  the 
introduction  of  Cantor's  transfinite  cardinal  numbers,  need  not 
concern  us  here;  the  definitions  of  the  principal  terms  which  are 
used  in  this  theory  will  be  found  in  chapter  VII. 

11.  After  the  theory  of  classes,  as  such,  which  is  logically  the 
simplest  branch  of  mathematics,  the  next  in  order  of  complexity  is 
the  theory  of  classes  in  which  a  relation  or  an  operation  among  the 
elements  is  defined.  For  example,  in  the  class  of  numbers  we  have 
the  relation  of  "  less  than  "  and  the  operations  of  addition  and 
multiplication;*  in  the  class  of  points,  the  relation  of  coUinearity, 
etc.;  in  the  class  of  human  beings,  the  relations  "  brother  of," 
"  debtor  of,"  etc. 

If  we  use  the  term  system  to  denote  a  class  together  with  any 
relations  or  operations  which  may  be  defined  among  its  elements 
we  may  say  that  the  simplest  mathematical  systems  are : 

(1)  a  class  with  a  single  relation,  and 

(2)  a  class  with  a  single  operation. 

The  most  important  example  of  the  first  kind  is  the  theory  of 
simply  ordered  classes,  which  forms  the  subject  of  the  present 
paper;  the  most  important  example  of  the  second  kind  is  the  theory 
of  abstract  groups.f  The  ordinary  algebra  of  real  or  complex 
numbers  is  a  combination  of  the  two.| 

*  As  M.  B6cher  has  pointed  out  [Bull.  Amer.  Math.  Soc,  vol.  11  (1904), 
p.  126],  any  operation  or  rule  of  combination  by  which  two  elements  determine 
a  third  may  be  interpreted  as  a  triadic  relation;  for  example,  instead  of  saying 
that  two  given  numbers  a  and  h  determine  a  third  number  c  called  their  sum 
(o  +  6  =  c),  we  may  say  that  the  three  elements  a,  b,  and  c  satisfy  a  certain 
relation  R  (a,  b,  c). 

t  For  a  bibhographical  account  of  the  definitions  of  an  abstract  group,  see 
Trans.  Amer.  Math.  Soc,  vol.  6  (1905),  pp.  181-193. 

X  For  a  definition  of  ordinary  algebra  by  a  set  of  independent  postulates,  see 
Trans.  Amer.  Math.  Soc,  vol.  6  (1905),  pp.  209-229,  or  my  two  monographs 
cited  in  the  introduction.    For  a  similar  definition  of  the  Boolean  algebra  of 


§  11  CLASSES  IN  GENERAL  9 

We  proceed  in  the  next  chapter  to  explain  the  conditions  or 
"  postulates  "  which  a  class,  K,  and  a  relation,  <  (or  "  /^  "),  must 
satisfy  in  order  that  the  system  (K,  <)  may  be  called  a  simply 
ordered  class. 

logic,  see  Trans.  Amer.  Math.  Soc,  vol.  5  (1904),  pp.  288-309  [compare  a 
recent  note  by  B.  A.  Bernstein,  Bull.  Amer.  Math.  Soc,  vol.  22  (1916),  pp.  458- 
459];  also  papers  by  H.  M.  Sheffer,  Trans.  Amer.  Math.  Soc,  vol.  14  (1913), 
pp.  481-488,  and  B.  A.  Bernstein,  Univ.  of  California  Publications  in  Math., 
vol.  1  (1914),  pp.  87-96,  and  Trans.  Amer.  Math.  Soc,  vol.  17  (1916),  pp.  50- 
62. 


CHAPTER  II 

General  Properties  of  Simply  Ordered  Classes 
OR  Series 

12.  If  a  class,  K,  and  a  relation,  -<  (called  the  relation  of  order), 
satisfy  the  conditions  expressed  in  postulates  0,  1-3,  below,  then 
the  system  (K,  <)  is  called  a  simply  ordered  class,  or  a  series* 
The  notation  a  <  6  or  (6  >  a,  which  means  the  same  thing),  may 
be  read:  "  a  precedes  b  "  (or  "  b  follows  a  ").  The  class  K  is  said 
to  be  arranged,  or  set  in  order,  by  the  relation  -< ,  and  the  relation 
<  is  called  a  serial  relation  within  the  class  K. 

Postulate  0.  The  class  K  is  not  an  empty  class,  nor  a  class  con- 
taining merely  a  single  element. 

This  postulate  is  intended  to  exclude  obviously  trivial  cases,  and 
will  be  assumed  without  further  mention  throughout  the  paper. 

Postulate  1.  If  a  and  b  are  distinct  elements  of  K,  then  either 
a  <  b  or  b  <  a.\ 

Postulate  2.   If  a  <  b,  then  a  and  b  are  distinct.X 
Postulate  3.   If  a  <  b  and  b  <  c,  then  a  <  c.§ 
The  consistency  and  independence  of  these  postulates  will  be 
established  in  §  19  and  §  20. 

13.  As  an  immediate  consequence  of  postulates  2  and  3,  we 
have 

Theorem  I.   If  a  <  b  is  true,  then  b  <  a  isfalse.\\ 

*  "  Einfach  geordnete  Menge:  "  G.  Cantor,  Math.  Ann.,  vol.  46  (1895), 
p.  496;  "  series:  "  B.  Russell,  Principles  of  Mathematics,  vol.  1  (1903),  p.  199. 

t  This  postulate  1  has  been  called  by  Russell  the  postulate  of  connexity; 
loo.  cit.,  p.  239. 

I  Any  relation  -<  which  satisfies  postulate  2  is  said  to  be  irreflexive 
throughout  the  class;  this  term  is  due  to  Peano;  see  Russell,  loc.  cit.,  p.  219. 

§  Any  relation  ■<  which  satisfies  postulate  3  is  said  to  be  transitive  through- 
out the  class.    This  term  has  been  in  common  use  since  the  time  of  DeMorgan. 

II  Any  relation  -<  which  has  this  property  is  said  to  be  asymmetrical  through- 
out the  class;  see  Russell,  loc.  cit.,  p.  218. 

10 


§  16  SIMPLY  ORDERED  CLASSES  OR  SERIES  11 

(For,  if  a  <  6  and  b  <  a  were  both  true,  we  should  have,  by  3, 
a  <  a,  whence,  by  2,  a  ^  a,  which  is  absurd). 

If  desired,  this  theorem  I  may  be  used  in  place  of  postulate  2  in 
the  definition  of  a  serial  relation. 

14.  The  general  properties  of  series  may  now  be  summarized  as 
follows : 

//  a  and  b  are  any  elements  of  K,  then  either 
a  =  b,  or  a  <  b,  or  b  <  a^ 

and  these  three  conditions  are  mutually  exclusive;  jurther,  if  a  <  b 
and  b  <  c,  then  a  <  c. 

These  are  the  properties  which  characterize  a  serial  relation 
within  the  class  K* 

15.  As  the  most  familiar  examples  of  series  we  mention  the 
following:  (1)  the  class  of  all  the  natural  numbers  (or  the  first  n  of 
them),  arranged  in  the  usual  order;  and  (2)  the  class  of  all  the 
points  on  a  line,  the  relation  "a  <  6  "  signifying  "  a  on  the  left 
of  6."  Many  other  examples  will  occur  in  the  course  of  our 
work. 

16.  If  two  series  can  be  brought  into  one-to-one  correspondence 
in  such  a  way  that  the  order  of  any  two  elements  in  one  is  the  same 
as  the  order  of  the  corresponding  elements  in  the  other,  then  the 
two  series  are  said  to  be  ordinally  similar,  or  to  belong  to  the  same 
type  of  order  (Ordnungstypus)  .'\ 

For  example,  the  class  of  all  the  natural  numbers,  arranged  in 
the  usual  order,  is  ordinally  similar  to  the  class  of  all  the  even 
numbers,  arranged  in  the  usual  order  (compare  §  3). 

Again,  the  class  of  all  the  points  on  a  line  one  inch  long,  arranged 
from  left  to  right,  is  ordinally  similar  to  the  class  of  all  the  points 
on  a  line  three  inches  long,  arranged  from  left  to  right  (compare 
§8). 

*  A  serial  relation  may  also  be  described  as  one  which  is  (1')  connected; 
(2')  irreflexive;  (3')  transitive  for  distinct  elements;  and  (4')  asymmetrical 
for  distil  ct  elements;  these  four  properties  [(3')  and  (4')  being  weaker  forms 
of  postulate  3  and  theorem  I  respectively]  are  readily  shown  to  be  independent. 
See  a  forthcoming  paper  by  E.  V.  Huntington  cited  in  §  20,  below. 

t  Cantor,  Math.  Ann.,  vol.  46  (1895),  p.  497. 


12  TYPES  OF  SERIAL  ORDER  §  17 

It  will  be  noticed  that  in  the  first  of  these  examples  the  corre- 
spondence between  the  two  series  can  be  set  up  in  only  one  way, 
while  in  the  second  example,  the  correspondence  can  be  set  up  in  an 
infinite  number  of  ways^  This  distinction  is  an  important  one,  for 
which,  unfortunately,  no  satisfactory  terminology  has  yet  been 
proposed.* 

17.  Before  giving  further  examples  of  the  various  types  of 
simply  ordered  classes,  it  will  be  convenient  to  give  here  the  defi- 
nitions of  a  few  useful  technical  terms. 

Definition  1.  In  any  series,  ii  a  <  x  and  x  <  h,  then  x  is  said 
to  lie  between  a  and  6.t 

Definition  2.  In  any  series,  if  a  <  x  and  no  element  exists 
between  a  and  x,  then  x  is  called  the  element  next  following  a,  or  the 
(immediate)  successor  of  a.  Similarly,  ii  y  <  a  and  no  element 
exists  between  y  and  a,  then  y  is  called  the  element  next  preceding  a, 
or  the  (immediate)  predecessor  of  a.J 

For  example,  in  the  class  of  natural  numbers  in  the  usual  order 
every  element  has  a  successor,  and  every  element  except  the  first 
has  a  predecessor;  but  in  the  class  of  points  on  a  fine,  in  the  usual 
order,  every  two  points  have  other  points  between  them,  so  that 
no  point  has  either  a  successor  or  a  predecessor. 

Definition  3.  In  any  series,  if  one  element  x  precedes  all  the 
other  elements,  then  this  x  is  called  the  first  element  of  the  series. 
Similarly,  if  one  element  y  follows  aU  the  others,  then  this  y  is 
called  the  last  element. 

18.  With  regard  to  the  existence  of  first  and  last  elements,  all 
series  may  be  divided  into  four  groups:  (1)  those  that  have  neither 
a  first  element  nor  a  last  element;  (2)  those  that  have  a  first  ele- 
ment, but  no  last;  (3)  those  that  have  a  last  element,  but  no  first; 
and  (4)  those  that  have  both  a  first  and  a  last. 

*  Cf.  Trans.  Amer.  Math.  Soc,  vol.  6  (1905),  p.  41;  or  O.  Veblen,  Bull. 
Amer.  Math.  Soc,  vol.  12  (1906),  p.  303.  One  might  speak  of  a  determinate 
correspondence  and  an  indeterminate  correspondence  (Bricard). 

t  For  an  elaborate  analysis  of  this  concept,  see  a  forthcoming  paper  called 
"  Sets  of  independent  postulates  for  betweenness,"  by  E.  V.  Huntington  and 
J.  R.  Kline,  Trans.  Amer.  Math.  Soc. 

J  See  footnote  t  under  §  31. 


§  19  SIMPLY  ORDERED  CLASSES  OR  SERIES  13 

For  example,  the  class  of  all  the  points  on  a  line  between  A  and  B, 

arranged  from  A  to  B,  has  no  first  point,    1)   A   B 

and  no  last  point,  since  if  any  point  C  of    2)   A    • B 

the  class  be  chosen  there  will  be  points  of    3)   A   •    B 

the  class  between  C  and  A  and  also  be-    4)   A    • •    B 

tween  C  and  B.  If,  however,  we  consider  a  new  class,  comprising  all 
the  points  between  A  and  B,  and  also  the  point  A  (or  B,  or  both), 
arranged  from  A  to  B,  then  this  new  class  will  have  a  first  element 
(or  a  last  element,  or  both).  The  four  cases  are  represented  in  the 
accompanying  diagram. 

Examples  of  series 

19.  In  this  section  we  give  some  miscellaneous  examples  of 
simply  ordered  classes,  to  illustrate  some  of  the  more  important 
types  of  serial  order.  Most  of  these  examples  will  be  discussed  at 
length  in  later  chapters. 

In  each  case  a  class  K  and  a  relation  <  are  so  defined  that  the 
system  (K,  < )  satisfies  the  conditions  expressed  in  postulates  1-3 
(§  12).  The  existence  of  any  one  of  these  systems  is  sufficient  to 
show  that  the  postulates  are  consistent,  that  is,  that  no  two  con- 
tradictory propositions  can  be  deduced  from  them.  For,  the 
postulates  and  all  their  logical  consequences  express  properties  of 
these  systems,  and  no  really  existent  system  can  have  contradictory 
properties.* 

(1)  K  =  the  class  of  all  the  natural  numbers  (or  the  first  n  of 
them),  with  <  defined  as  "  less  than." 

This  is  an  example  of  a  "  discrete  series  "  (see  chapter  III). 

(2)  K  =  the  class  of  all  the  points  on  a  line  (with  or  without 
end-points),  with  <  defined  as  "  on  the  left  of." 

This  is  an  example  of  a  "  continuous  series  "  (see  chapter  V). 

*  On  the  consistency  of  a  set  of  postulates,  see  a  problem  of  D.  Hilbert's, 
translated  in  Bull.  Amer.  Math.  Soc,  vol.  8  (1902),  p.  447,  and  a  paper  by 
A.  Padoa,  L' Enseignement  Mathematique,  vol.  5  (1903),  pp.  85-91.  Also  D. 
Hilbert,  Verhandl.  des.  3.  internat.  Math.-Kongresses  in  Heidelberg,  1904,  pp. 
17-^185;  French  translation,  Ens.  Math.,  vol.  7  (1905),  pp.  89-103;  EngUsh 
translation,  Monist,  vol.  15  (1905),  pp.  338-352. 


14  TYPES  OF  SERIAL  ORDER  §  19 

(3)  K  =  the  class  of  all  the  points  on  a  square  (with  or  without 
the  points  on  the  boundary),  with  -<  defined  as  follows:  let  x  and 
y  represent  the  distances  of  any  point  of  the  square  from  two  adja- 
cent sides;  then  of  two  points  which  have  unequal  a:'s,  the  one 
having  the  smaller  x  shall  precede,  and  of  two  points  which  have 
the  same  x,  the  one  having  the  smaller  y  shall  precede.  In  this  way 
all  the  points  of  the  square  are  arranged  as  a  simply  ordered  class. 

(4)  By  a  similar  device,  the  points  of  all  space  can  be  arranged  as 
a  simply  ordered  class.  Thus,  let  x,  y,  and  z  be  the  distances  of 
any  point  from  three  fixed  planes;  then  in  each  of  the  eight  octants 
into  which  all  space  is  divided  by  the  three  planes,  arrange  the 
points  in  order  of  magnitude  of  the  x's,  or  in  case  of  equal  x's,  in 
order  of  magnitude  of  the  y^s,  or  in  case  of  equal  rr's  and  equal  y's, 
in  order  of  magnitude  of  the  z's;  and  finally  arrange  the  octants 
themselves  in  order  from  1  up  to  8,  paying  proper  attention  to  the 
points  on  the  bounding  planes. 

(5)  K  =  the  class  of  all  proper  fractions,  arranged  in  the  usual 
order. 

This  is  an  example  of  a  series  called  "  denumerable  and  dense  " 
(see  chapter  IV). 

By  a  proper  fraction  (written  m/n)  we  mean  an  ordered  pair  of 
natural  numbers,  of  which  the  first  number,  m,  called  the  numera- 
tor, and  the  second  number,  n,  called  the  denominator,  are  rela- 
tively prime,  and  m  is  less  than  n;  and  by  the  "  usual  order  "  we 
mean  that  a  fraction  7n/n  is  to  precede  another  fraction  p/q  when- 
ever the  product  m  X  ^  is  less  than  the  product  n  X  p.  The  class 
as  so  ordered  clearly  satisfies  the  conditions  1-3,  as  one  sees  by  a 
moment's  calculation. 

(6)  K  =  the  class  of  all  proper  fractions  arranged  in  a  special 
order,  as  follows :  of  two  fractions  which  have  unequal  denomina- 
tors, the  one  having  the  smaller  denominator  shall  precede,  and  of 
two  fractions  which  have  the  same  denominator  the  one  having  the 
smaller  numerator  shall  precede. 

In  contrast  with  example  (5),  this  series  is  of  the  same  type  as  the 
series  of  the  natural  numbers  arranged  in  the  usual  order,  as  the 
following  correspondence  will  show  (compare  §  42)  :* 
*  Cf.  G.  Cantor,  loc.  cit.  (1895),  p.  496. 


§  19  SIMPLY  ORDERED  CLASSES  OR  SERIES  15 


1 

2     3 

4     5 

6     7    8     9 

10 

11        •  • 

1 

1     2 

1     3 

12     3     4 

1 

5 

2 

3     3 

4    4 

5     5     5     5 

6 

6 

These  two  examples,  (5)  and  (6),  illustrate  the  obvious  fact  that 
the  same  class  may  be  capable  of  being  arranged  in  various  different 
orders. 

(7)  As  another  example,  let  X  be  a  class  whose  elements  are 
natural  numbers  affected  with  other  natural  numbers  as  subscripts; 
for  example,  li,  54,  etc.;  and  let  the  relation  of  order  be  defined  as 
follows:  of  two  numbers  which  have  unequal  subscripts,  the  one 
having  the  smaller  subscript  shall  precede,  and  of  two  numbers 
which  have  the  same  subscript,  the  smaller  number  shall  precede. 
The  system  may  be  represented  thus,  the  relation  <  being  read  as 
"  on  the  left  of:  " 

li,  2i,  3i,  .  .  .;  I2,  22,  32,  .  .  .;  I3,  23,  33,  ...;...  . 
This  is  an  example  of  what  Cantor  has  called,  in  a  technical 
sense,  a  "  well-ordered  series  "  (see  chapter  VII). 

(8)  An  example  of  a  somewhat  different  character  is  the  follow- 
ing: *  let  K  be  the  class  of  all  possible  infinite  classes  of  the  natural 
numbers,  no  number  being  repeated  in  any  one  class;  f  and  let 
these  classes  be  arranged,  or  set  in  order,  as  follows:  any  class  a 
shall  precede  another  class  6  when  the  smallest  number  in  a  is  less 
than  the  smallest  number  in  h,  or,  if  the  smallest  n  numbers  of 
a  and  b  are  the  same,  when  the  (w  +  l)st  number  of  a  is  less  than 
the  (n  -|-  l)st  number  of  b. 

A  moment's  reflection  shows  that  this  system  satisfies  the  condi- 
tions for  an  ordered  class;  it  will  appear  later  that  it  belongs  to  the 
type  of  series  called  continuous  (see  §  63,  5). 

A  more  familiar  example  of  the  same  type  is  the  following: 

(9)  K  =  the  class  of  all  non-terminating  decimal  fractions  be- 
tween 0  and  1,  arranged  in  the  usual  order.     (Compare  §  40.) 

*  B.  Russell,  Principles  of  Mathematics,  vol.  1,  p.  299. 

t  For  example,  the  class  of  all  prime  numbers,  or  the  class  of  all  even  num- 
bers, or  the  class  of  all  even  numbers  greater  than  1000,  or  the  class  of  all  perfect 
cube  nvunbers,  or  the  class  of  all  numbers  that  begin  with  9,  or  the  class  of  all 
numbers  that  do  not  contain  the  digit  5,  would  be  an  element  of  K. 


16  TYPES  OF  SERIAL  ORDER  §20 

By  a  non-terminating  decimal  fraction  between  0  and  1,  we  mean 
a  rule  or  agreement  by  which  every  natural  number  has  assigned  to 
it  some  one  of  the  ten  digits  0,  1,  2,  .  .  .  ,  9,  excluding,  however, 
the  rules  which  would  assign  a  0  to  every  number  after  any  given 
number  (these  excluded  rules  giving  rise  to  the  terminating  deci- 
mals).* The  digit  assigned  to  any  particular  number  n  is  called  the 
nth  digit  of  the  decimal,  or  the  digit  in  the  nth  place.  By  the 
"  usual  order  "  within  this  class,  we  mean  that  any  decimal  a  is  to 
precede  another  decimal  b  when  the  first  digit  of  a  is  less  than  the 
first  digit  of  6,  or,  if  the  first  n  digits  of  a  and  b  are  the  same,  when 
the  (n  +  l)st  digit  of  a  is  less  than  the  (n  +  l)st  digit  of  6  (the 
digits  being  taken  in  the  order  of  magnitude  from  0  to  9). 

All  these  examples  of  simply  ordered  classes  have  been  chosen 
from  the  domains  of  arithmetic  and  geometry;  among  the  other 
examples  which  readily  suggest  themselves  the  following  may  be 
mentioned : 

(10)  The  class  of  all  instants  of  time,  arranged  in  order  of 
priority. 

(11)  The  class  of  all  one's  distinct  sensations,  of  any  particular 
kind,  as  of  pleasure,  pain,  color,  warmth,  sound,  etc.,  arranged  in 
order  of  intensity. 

(12)  The  class  of  all  events  in  any  causal  chain,  arranged  in  order 
of  cause  and  effect. 

(13)  The  class  of  all  moral  or  commercial  values,  arranged  in 
order  of  superiority. 

(14)  The  class  of  all  measurable  magnitudes  of  any  particular 
kind,  as  lengths,  weights,  volumes,  etc.,  arranged  in  order  of  size. 

Examples  of  systems  (K,  < )  which  are  not  series 

20.  In  this  section  we  give  some  examples  of  systems  (K,  < ) 
which  are  not  series  because  they  satisfy  only  two  of  the  three  con- 
ditions expressed  in  postulates  1-3  (§  12).  The  existence  of  these 
systems  proves  that  the  three  postulates  are  independent  —  that 
is,  that  no  one  of  them  can  be  deduced  from  the  other  two.    (For, 

*  It  should  be  noticed  that  what  we  are  here  required  to  grasp  is  not  the 
infinite  totahty  of  digits  in  the  decimal  fraction,  but  simply  the  nile  by  which 
those  digits  are  determined. 


§20  SIMPLY  ORDERED  CLASSES  OR  SERIES  17 

if  any  one  of  the  three  properties  were  a  logical  consequence  of  the 
other  two,  every  system  which  had  the  first  two  properties  would 
have  the  third  property  also,  which,  as  these  examples  show,  is  not 
the  case.)  In  other  words,  no  one  of  the  three  postulates  is  a  re- 
dundant part  of  the  definition  of  a  serial  relation.* 

(1)  Systems  not  satisfying  postulate  1  (namely:  if  a  9^  b,  then 
a<borb<a). 

(a)  Let  K  be  the  class  of  all  natural  numbers,  with  <  so  defined 
that  a  precedes  b  when  and  only  when  2a  is  less  than  b. 

(6)  Let  K  be  the  class  of  all  human  beings,  throughout  history, 
with  <  defined  as  "  ancestor  of." 

(c)  Let  K  be  the  class  of  all  points  {x,  y)  in  a  given  square,  with 
(xi,  yi)  <  (x2,  2/2)  when  and  only  when  Xi  is  less  than  X2  and  yi  less 
than  ^2. 

In  all  these  systems,  postulates  2  and  3  are  clearly  satisfied. f 

(2)  Systems  not  satisfying  postulate  2  (namely:  if  a  <  6,  then 
a^  b). 

(a)  Let  K  be  the  class  of  all  natural  numbers  with  a  <  b  signify- 
ing "  a  less  than  or  equal  to  6." 

(6)  Let  K  be  any  class,  with  a  <  b  signifying  "  a  is  co-existent 
with  6." 

Both  these  systems  satisfy  postulates  1  and  3. 

(3)  Systems  not  satisfying  postulate  3  (namely:  if  a  <  6  and 
b  <  c,  then  a  <  c). 

(a)  Let  K  be  the  class  of  all  natural  numbers,  with  <  meaning 
"  different  from." 

*  This  method  of  proving  the  independence  of  a  set  of  postulates  is  the 
method  which  has  been  made  familiar  in  recent  years  by  the  work  of  Peano 
(1889),  Padoa,  Fieri,  and  Hilbert  (1899).  For  a  discussion  of  the  "  complete 
independence  "  of  these  postulates  in  the  sense  defined  by  E.  H.  Moore  (1910), 
see  a  forthcoming  paper  by  E.  V.  Huntington,  Complete  existential  theory  of  the 
postulates  for  serial  order,  Bull.  Amer.  Math.  Soc.  (1917). 

t  Another  very  interesting  example  of  a  system  of  this  kind  is  the  so- 
called  "  conical  order  "  studied  by  A.  A.  Robb  in  his  book:  A  Theory  of  Time 
and  Space  (Cambridge,  Eng.,  1914). 


18  TYPES  OF  SERIAL  ORDER  §20 

(6)  Let  X  be  a  class  of  any  odd  number  of  points  distributed  at 
equal  distances  around  the  circumference  of  a  circle,  with  a  <  b 
meaning  that  the  arc  from  a  to  h,  in  the  counter-clockwise  direction 
of  rotation,  is  less  than  a  semi-circle.    - 

(c)  Let  X  be  a  family  of  brothers,  with  a  <  b  signifying  "  a  is  a 
brother  of  6."  This  relation  is  not  transitive,  since  from  a  <  b  and 
6  -<  a  it  does  not  follow  that  a  <  a. 

All  three  of  these  systems  clearly  satisfy  postulates  1  and  2. 

In  the  following  chapters  we  consider  in  detail  those  types  of 
series  which  are  especially  important  in  the  study  of  algebra. 


CHAPTER  III 

Discrete  Series:  Especially  the  Type  oj  of  the 
Natural  Numbers 

21.  A  discrete  series  may  be  defined  as  any  series  {K,  < )  which 
satisfies  not  only  the  general  conditions  1-3  of  §  12,  but  also  the 
special  conditions  expressed  in  postulates  N1-N3,  below: 

Postulate  A'"1.  (Dedekind's  postulate*)  If  Ki  and  K^  are  any 
two  non-empty  parts  of  K,  such  that  every  element  of  K  belongs 
either  to  Ki  or  to  K^  and  every  element  of  Ki  precedes  every  element  of 
Kz,  then  there  is  at  least  one  element  X  in  K  such  that: 

(1)  any  element  that  precedes  X  belongs  to  Ki,  and 

(2)  any  element  that  follows  X  belongs  to  K^. 

The  significance  of  this  postulate  A''!  will  be  best  explained  by 
the  examples,  given  below,  of  series  which  have  and  those  which  do 
not  have  the  property  in  question.  For  the  present  it  is  sufficient 
to  remark  that  whenever  the  postulate  is  satisfied,  Ki  will  have  a 
last  element,  or  K^  will  have  a  first  element,  or  both;  whichever 
one  of  these  elements  exists  (or  either  of  them  if  they  both  exist) 
will  serve  as  the  element  X  required,  and  may  be  said  to  "divide" 
the  two  parts  Ki  and  Ki. 

Postulate  A^2.  Every  element  of  K,  unless  it  be  the  last,  has  an 
immediate  successor  (§  17). 

Postulate  NS.  Every  element  of  K,  unless  it  be  the  first,  has  an 
immediate  predecessor  (§  17). 

The  consistency  and  independence  of  these  postulates  are  shown 
in  §§  28-29. 

*  R.  Dedekind,  Stetigkeit  und  irrationale  Zahlen,  1872;  cf.  §  62,  below.  The 
selection  of  postulates  here  given  for  discrete  series  is  the  same  as  that  adopted 
by  O.  Veblen,  Trans.  Amer.  Math.  Soc,  vol.  6  (1905),  pp.  165-171.  As  far  as  I 
know,  Dedekind's  postulate  had  not  been  used  by  earlier  writers  in  this  con- 
nection. 

19 


20  TYPES  OF  SERIAL  ORDER  §22 

22.  An  example  of  a  discrete  series  is  the  class  of  all  integers 
(positive,  negative,  and  zero),  arranged  in  the  usual  order: 

.  .  .,  -3,  -2,  -1,    0,     +1,  +2,  +3,  . .  .  . 

The  elements  of  this  system  are  of  three  kinds :  (1)  the  positive 
integers,  which  are  natural  numbers  affected  with  the  sign  + ;  (2) 
the  negative  integers,  which  are  natural  numbers  affected  with  the 
sign  —  ;  and  (3)  an  extra  element  called  zero.  The  *'  usual  order  " 
is  more  precisely  defined  as  follows:  of  two  positive  integers,  the 
one  that  is  numerically  smaller  precedes;  of  two  negative  integers, 
the  one  that  is  numerically  greater  precedes;  every  negative  in- 
teger precedes  and  every  positive  integer  follows  the  integer  zero; 
and  of  two  integers  of  opposite  signs,  the  negative  precedes  the 
positive. 

By  making  this  series  terminate  in  one  or  both  directions  we 
have  an  example  of  a  discrete  series  with  a  first  element  or  a  last 
element  or  both.     (For  another  example,  see  §  28.) 

23.  The  most  important  property  of  discrete  series  is  expressed 
in  the  often  cited  "  theorem  of  mathematical  induction,"  which 
may  be  stated  in  the  following  form : 

Theorem  of  mathematical  induction.  If  a  and  6  are  any  two  ele- 
ments of  a  discrete  series,  and  a  <  b,  then :  if  we  start  from  a  and 
form  the  sequence  of  elements  pi,  p^,  ps,  .  .  .  ,  in  which  pi  is  the 
successor  of  a,  p2  the  successor  of  pi,  and  so  on,  some  one  of  these  p's 
will  be  the  element  b;  or  again,  if  we  start  from  b  and  form  the 
sequence  qi,  q^,  gs,  .  .  .  ,  in  which  qi  is  the  predecessor  of  6,  qi  the 
predecessor  of  gi,  and  so  on,  some  one  of  these  q's  will  be  the  element  a. 

In  other  words,  the  class  of  elements  between  any  two  elements  of 
a  discrete  series  can  be  exhausted  by  taking  away  its  elements  one 
by  one,  and  is  therefore  a  finite  class  (by  §  10). 

The  significance  of  this  theorem  will  be  clearer  after  a  study  of 
the  examples  in  §  29  of  series  in  which  the  theorem  does  not  hold. 
The  formal  proof  from  postulates  1-3  and  N1-N3  is  as  follows: 

Suppose,  in  the  first  case  considered  in  the  theorem,  that  the 
sequence  a,  pi,  P2,  ps,  •  -  -  (which  we  shall  call  the  sequence  P) 
did  not  contain  the  element  b.  On  this  supposition,  b  would  come 
after  all  the  elements  of  P,  and  we  could  divide  the  whole  series  K 
into  two  non-empty  parts,  namely:  Ki,  containing  every  element 


§24  DISCRETE  SERIES  21 

which  is  equalled  or  surpassed  by  any  element  of  P;  and  K2,  con- 
taining every  element  which  (like  the  element  b)  comes  after  all 
the  elements  of  P.  Then  by  Dedekind's  postulate  there  would  be 
an  element  X  "  dividing  "  Ki  from  K2  so  that  the  predecessor  of  X 
would  belong  to  P  while  the  successor  of  X  would  not.  But  this  is 
impossible,  since,  by  the  way  in  which  the  sequence  P  is  con- 
structed, if  the  predecessor  of  X  belonged  to  P,  then  X  itself,  and 
hence  the  successor  of  X,  would  also  belong  to  P.  Thus  the  sup- 
position with  which  we  started  has  led  to  contradiction,  and  the 
first  half  of  the  theorem  is  proved. 

The  second  half  is  proved  in  a  similar  way. 

All  discrete  series  may  be  divided  into  four  groups,  distinguished 
by  the  presence  or  absence  of  extreme  elements;  we  consider  the 
four  cases  separately,  as  follows : 

1.   Progressions:  series  of  the  type  "  co." 

24.  A  discrete  series  (§  21)  which  has  a  first  element,  but  no  last, 
is  called  a  progression* 

All  progressions  are  ordinally  similar,  that  is,  any  two  of  them 
can  be  brought  into  one-to-one  correspondence  in  a  way  that  pre- 
serves the  relations  of  order. 

For,  we  can  assign  the  first  element  of  one  of  the  progressions  to 
the  first  element  of  the  other,  the  successor  of  that  element  in  one 
to  the  successor  of  that  element  in  the  other,  and  so  on ;  and  by  the 
theorem  of  mathematical  induction  no  element  of  either  series  will 
be  inaccessible  to  this  process. 

We  may  therefore  speak  of  the  progressions  as  constituting  a 
definite  type  of  order,  which  Cantor  f  has  called  the  type  co.  More- 
over, the  ordinal  correspondence  between  two  progressions  can  be 
set  up  in  only  one  way;  this  fact  will  be  useful  to  us  later  (see  §  31). 

The  simplest  example  of  a  progression  is  the  series  of  natural 
numbers  in  the  usual  order: 

1,     2,     3,     .... 

Other  examples  are :  the  even  numbers,  or  the  prime  numbers,  or 
the  perfect  square  numbers,  in  the  usual  order;  or  the  proper  frac- 
tions arranged  in  the  special  order  described  in  §  19,  6. 

*  B.  Russell,  Principles  of  Mathematics,  vol.  1,  p.  239. 
t  G.  Cantor,  Math.  Ann.,  vol.  46  (1895),  p.  499. 


22  TYPES  OF  SERIAL  ORDER  §25 

2.   Regressions:  series  of  the  type  "  *o}." 

25.  A  discrete  series  (§21)  which  has  a  last  element  but  no  first 
is  called  a  regression. 

The  regressions,  like  the  progressions,  constitute  a  definite  type 
of  order,  which  Cantor  has  called  the  type  *co  (read:  star  omega). 
The  simplest  example  of  a  regression  is  the  series  of  negative 
integers  with  or  without  zero,  arranged  in  the  usual  order,  thus : 

.  .  .  ,  -3,  -2,  -1,  0. 

3.   Series  of  the  type  "  *co  +  co." 

26.  A  discrete  series  (§21)  which  has  neither  a  first  nor  a  last 
element  may  be  called  an  unlimited  discrete  series,  the  simplest 
example  being  the  series  of  all  integers  in  the  usual  order  (§  22). 

In  any  unlimited  discrete  series,  if  any  element  is  chosen  as  an 
"  origin,"  the  elements  preceding  this  element  form  a  regression 
and  those  following  it  a  progression;  hence  all  unlimited  discrete 
series  are  ordinally  similar,  and  constitute  a  third  definite  tj^pe  of 
order.  Cantor  denotes  this  type  by  *w  +  w,  the  plus  sign  being 
used  to  indicate  that  a  series  of  the  type  *co  is  to  be  followed  by  a 
series  of  the  type  co,  and  the  whole  regarded  as  a  single  series. 

It  should  be  noticed  that  the  correspondence  between  two  series 
of  the  type  *co  +  w  can  be  set  up  in  an  infinite  number  of  ways, 
since  any  element  may  be  taken  as  the  origin;  compare  the  follow- 
ing scheme: 

.  .  .  ,  -4,  -3,  -2,  -1,     0,  +1,  +2,  +3,  +4,  .  .  . 

.  .  .  ,  -2,  -1,     0,  +1,  +2,  +3,  +4,  +5,  +6, 

4.   Finite  series 

27.  A  discrete  series  (§  21)  which  has  a  first  element  and  a  last 
element  will  be  simply  a  finite  series,  the  word  finite  being  used  in 
the  sense  defined  in  §  7. 

For,  by  the  theorem  of  mathematical  induction  (§  23),  the  class 
of  elements  in  such  a  series  can  be  exhausted  by  taking  the  elements 
away  one  by  one;  therefore,  by  §  10,  it  cannot  be  an  infinite  class. 


§29  ,  .         DISCRETE  SERIES  23 

And  conversely,  every  finite  class  can  be  put  into  one-to-one  corre- 
spondence with  a  terminated  portion  of  a  discrete  series. 

These  theorems  may  be  used,  if  one  prefers,  as  the  definition  of  a 
finite  class  (compare  §  7) ;  an  infinite  class  would  then  be  defined 
as  one  which  is  not  finite. 

Other  examples  of  discrete  series 

28.  The  examples  of  a  discrete  series  so  far  mentioned  have  all 
been  drawn  from  the  domain  of  arithmetic  (as  the  series  of  all 
integers,  the  series  of  all  positive  integers,  the  series  of  all  negative 
integers,  and  series  containing  only  a  finite  number  of  elements). 
The  existence  of  any  one  of  these  systems  is  sufficient  to  establish 
the  consistency  of  the  postulates  of  this  chapter  (compare  §  19). 
In  this  section  we  give  a  non-numerical  example,  due  essentially  to 
Dedekind,  and  phrased  in  its  present  form  by  Royce:  * 

Suppose  a  complete  map  of  London  could  be  laid  out  on  the 
pavement  of  one  of  the  squares  of  the  city;  then  the  city  of  London 
would  be  represented  an  infinite  number  of  times  in  this  map,  and 
the  successive  representations  would  form  a  progression.  For  the 
map  itself  would  form  a  part  of  the  object  which  it  represents,  and 
would  therefore  include  a  miniature  representation  of  itself;  this 
representation  being  again  a  complete  map  of  the  city  would  con- 
tain a  still  smaller  representation  of  itself;  and  so  on,  ad  infinitum.^ 

Examples  of  series  which  are  not  discrete 

29.  In  this  section  we  give  some  examples  of  series  (§  12)  which 
are  not  discrete  (§21),  each  example  being  a  series  {K,  <)  which 
satisfies  two  of  the  postulates  Nl-NS  but  not  the  third.  The 
existence  of  these  systems  proves  (see  §  20)  that  the  postulates 
iVl-A^3  are  independent,  that  is,  that  no  one  of  them  is  redundant 
in  the  definition  of  a  discrete  series. 

*  R.  Dedekind,  Was  sind  und  was  sollen  die  Zahlen,  1887;  J.  Royce,  The 
World  and  the  Individual,  vol.  1,  1900,  p.  503. 

t  Another  example  of  such  a  self-representative  system  is  a  label  on  a  can  of 
baking-powder,  containing  a  picture  of  the  can.  Another  example  is  pro- 
vided by  the  images  observed  in  a  pair  of  parallel  mirrors. 


24  TYPES  OF  SERIAL  ORDER  §29 

(1)  A  system  not  satisfying  Nl  (Dedekind's  postulate).  Let  K 
consist  of  two  sets  of  integers  —  call  them  red  and  blue  —  the 
integers  of  each  set  being  positive,  negative,  or  zero;  and  let  the 
elements  be  arranged  along  a  line  from  left  to  right,  as  follows: 

red  blue 

'...,-2,-1,  o,+i,+2, ..:  '. . . , -2, -1,  0,  +1,  +2, . . . ; 

This  system  is  a  series  in  which  every  element  has  a  successor, 
and  every  element  has  a  predecessor;  but  Dedekind's  postulate, 
although  it  holds  in  general,  fails  in  case  Ki  contains  all  the  red 
elements  and  K^  all  the  blue. 

By  leaving  out  the  negative  integers  in  the  red  set,  or  the  positive 
integers  in  the  blue  set,  or  both,  we  can  readily  construct  a  series 
of  the  same  sort  having  either  or  both  extreme  elements;  the  series 
as  it  stands  has  neither. 

(2)  A  system  not  satisfying  N2  (on  successors).  Let  K  consist  of 
a  set  of  negative  integers  (in  red),  followed  by  a  set  of  all  integers 
(in  blue),  arranged  in  the  usual  order,  as  indicated  here: 

red  blue 

'.  •  • ,  -3,  -2,  -i;    .' . .  -2,  -1, 0,  +1,  +2,  +3, . . : 

In  this  series  every  element  has  a  predecessor,  and  Dedekind's 
postulate  is  satisfied  in  all  cases;  but  the  element  ~1  of  the  red  set 
has  no  immediate  successor. 

Systems  of  the  same  sort,  with  one  or  both  extreme  elements,  can 
be  at  once  derived. 

(3)  A  system  not  satisfying  NS  (on  predecessors).  Similarly,  let 
K  consist  of  a  set  of  all  integers  (in  red),  followed  by  a  set  of  positive 
integers  (in  blue),  arranged  as  follows: 

red  blue 

'. . . ,  -2,  -1,  0,  +1,  +2, . . :     +1,  +2,  +3, . . . : 

The  theorem  of  mathematical  induction  is  false  in  all  these  sys- 
tems, since  we  cannot  pass  from  a  red  element  to  a  blue  element  by 
a  finite  number  of  steps. 

Examples  of  series  which  satisfy  none  of  the  postulates  N1-N3 
will  occur  in  the  following  chapter  (§  51). 


§30  DISCRETE  SERIES  25 

Numbering  the  elements  of  a  discrete  series 
30.   By  "  numbering  "  the  elements  of  a  discrete  series,  we  mean 
simply  attaching  to  each  element  some  label  or  tag,  by  which  it  can 
be  permanently  recognized,  and  distinguished  from  any  other 
element. 

If  the  given  series  has  a  first  element  or  a  last  element  (or  both), 
this  may  be  accomplished  as  follows,  by  the  use  of  ten  characters 
called  digits,  1,  2,  3,  4,  5,  6,  7,  8,  9,  0. 

In  the  case  of  a  progression,  denote  the  first  element  by  1 ;  the 
successor  of  1  by  2 ;  the  successor  of  2  by  3 ;  and  so  on,  until  the 
successor  of  8  is  denoted  by  9.  Then  denote  the  successor  of  9  by  10 
(read  "  one,  zero  ") ;  the  successor  of  10  by  11  (read  "  one,  one  ") ; 
the  successor  of  11  by  12;  and  so  on,  until  the  successor  of  18  is 
denoted  by  19.  Then  denote  the  successor  of  19  by  20;  the  suc- 
cessor of  20  by  21;  and  so  on,  the  successor  of  99  being  denoted  by 
100,  etc. : 

1,  2,  3, 

By  carrying  the  process  far  enough  any  given  element  of  the  progres- 
sion can  be  reached,  in  virtue  of  the  theorem  of  mathematical 
induction. 

In  the  case  of  a  regression,  we  can  number  the  elements  in  a 
similar  way,  if  we  begin  with  the  last  element  and  run  backward. 
In  this  case  it  is  customary  to  attach  the  sign  -  to  each  label,  the 
last  element  of  the  series  being  denoted  by  ~1,  the  predecessor  of 
~1  by  "2,  the  predecessor  of  ~2  by  ~3,  and  so  on: 

.  .  .  ,  -3,  -2,  -1. 

In  the  case  of  a  finite  discrete  series,  the  elements  may  be  num- 
bered in  either  way,  forward  or  backward : 

1,     2,     3,     4,     5, 
-5,  -4,  -3,  -2,  -1. 

If,  however,  the  given  series  is  unlimited  (§  26),  there  is  no  ele- 
ment which  we  can  take  as  an  absolute  starting  point,  since  no 
element  is  distinguished  from  the  rest  by  any  ordinal  property. 
The  best  we  can  do  in  this  case  is  to  choose  arbitrarily  some  element 


26  TYPES  OF  SERIAL  ORDER  §31 

as  an  origin,  denoted  by  0,  and  then  number  the  elements  following 
0  as  a  progression,  and  the  elements  preceding  0  as  a  regression;  in 
this  way  each  element  has  attached  to  it  a  label  which  indicates  its 
position  in  the  series,  not  absolutely,  but  with  reference  to  the 
arbitrarily  chosen  origin: 

.  .  .  ,  -3,  -2,  -1,    0,  +1,  +2,  +3, 

It  should  be  noticed  in  all  these  cases  that  the  process  of  labelling 
the  elements  does  not  involve  the  notion  of  "  counting  "  in  the 
sense  of  ascertaining  "how  many";  the  combination  of  digits 
attached  to  each  element  is  simply  a  tag  by  which  it  can  be  recog- 
nized, like  the  numbers  in  a  telephone  book;  when  any  two 
elements  thus  labelled  are  given,  we  can  determine  at  once  which 
precedes  the  other  in  the  series  without  concerning  ourselves  at  all 
with  the  question  "  how  many  "  elements  may  lie  between  them.* 

Digression  on  sums  and  products  of  the  elements  of  a 
discrete  series 

31.  The  same  principle  of  mathematical  induction  which  made 
it  possible  to  "  number  "  each  element  of  a  discrete  series  (§  30), 
makes  it  possible  to  define  the  sum  and  the  product  of  any  two 
elements  of  such  a  series  in  terms  of  the  relation  of  order. f    If  the 

*  Instead  of  the  decimal  system  of  numeration  here  described  we  can  use 
also  the  less  familiar,  but  often  more  convenient,  binary  system,  in  which  only 
two  digits  are  required.  Thus,  in  the  binary  system  the  successive  elements  of 
a  progression  would  be  denoted  by:  1;  10,11;  100,101,110,111;  1000,1001, 
1010, 1011, 1100, 1101, 1110, 1111;  10000,  etc.  (The  digits  are  read  separately: 
101  =  "  one,  zero,  one,"  etc.)  The  advantage  of  any  such  system  of  numera- 
tion over  the  primitive  system  of  strokes  (/,  //,  ///,  ////,  etc.)  lies  in  the  fact 
that  each  digit  acquires  a  special  value  by  virtue  of  the  place  which  it  occupies  in 
the  symbol. 

t  The  following  sections  (§§32-35)  are  due  essentially  to  Peano  (1889), 
although  Peano's  postulates  for  a  progression  are  based  not  on  the  notion  of 
order,  but  on  the  notion  of  "  successor  of."  The  postulates  adopted  in  the 
present  paper  seem  to  me  preferable  in  several  respects  to  those  employed  by 
Peano,  especially  in  the  use  of  Dedekind's  postulate  in  place  of  the  more  obvious 
postulate  of  mathematical  induction  (cf.  footnote  under  §  21).  A  brief 
account  of  Peano's  postulates  wiU  be  found  in  Bull.  Amer.  Math.  Soc,  vol.  9 


§33  DISCRETE  SERIES  27 

series  has  a  first  element  or  a  last  element  (or  both),  the  sums  and 
products  are  defined  absolutely ;  if  the  series  is  unlimited,  the  sums 
and  products  are  defined  with  reference  to  an  arbitrarily  chosen 
origin, 

32.  We  begin  with  the  general  case  of  an  unlimited  discrete 
series,  and  suppose  that  an  origin  has  been  chosen  and  the  elements 
labelled  as  in  the  preceding  section :  ^ 

.  .  .  ,  -3,  -2,  -1,     0,  +1,  +2,  +3, ' 

The  sum,  a  -\-  b  oi  two  elements  a  and  b,  with  respect  to  the 
origin  0,  is  then  defined  as  follows : 

(1)  a  +  0  =  a  and  0  +  a  =  a. 

(2)  a  +  +1  =  the  successor  of  a;  a  -\-  +2  =  the  successor  of 
a  +  +1;  a  -{-  +S  =  the  successor  of  a  ++2;  and  so  on;  in  general, 

a  +  (the  successor  of  +n)  =  the  successor  of  (a  +  +n). 

(3)  a  +  ~1  =  the  predecessor  of  a;  a  +  ~2  =  the  predecessor 
ofa  +  ~l;  a  +  "3  =  the  predecessor  of  a  +  ~2;  and  so  on;  in 
general, 

a  +  (the  predecessor  of  ~n)  =  the  predecessor  of  (a  +  ~w)  • 
In  this  way  the  sura  of  any  two  elements  can  be  determined,  by 

virtue  of  the  theorem  of  mathematical  induction  (§  23). 

On  the  basis  of  this  definition  of  the  sum,  the  product  a  X  b 

(or  a  .  6,  or  ab)  of  a  and  b,  with  respect  to  the  origin  0,  is  defined  as 

follows : 

(1)  0  X  a  =  0  and  a  X  0  =  0. 

(2)  +1  X  a  =  a;  +2  X  a  =  (+la)  +  a;  +3  X  a  =  (+2a)  +  a; 
and  so  on ;  in  general  (the  successor  of  +n)  X  a  =  ('^na)  +  a. 

(3)  ~n  X  a  =  '^n  X  a  with  its  sign  reversed. 

By  these  rules  the  product  of  any  two  elements  can  be  deter- 
mined. 

33.  From  these  definitions  the  following  fundamental  theorems  * 
can  be  readily  established : 

(1902),  p.  41,  and  an  extended  discussion  in  Russell,  loc.  cit.,  chap.  14.  A  re- 
vised list,  in  which  the  number  of  postulates  is  reduced  to  four,  is  given  by 
A.  Padoa,  Rev.  de  Math.  vol.  8  (1902),  p.  48. 

*  See  my  two  monographs  cited  in  the  introduction. 


f  ^^  K^      -  C^-^t)   -     -  %  t  - 


28  TYPES  OF  SERIAL  ORDER  §34 

(1)  (a  -\-  h)  -{-  c  =  a  -{-  {b  -{-  c) .    (Associative  law  for  addition.) 

(2)  a  +  6  =  6  +  a.     (Commutative  law  for  addition.) 

{  r\  ^  {ab)c  =  a(hc).     (Associative  law  for  multiplication.) 

(4)  ab  =  ba.     (Commutative  law  for  multiplication.) 
/  si  (5)  a(b  -{-  c)  =  ab  -\-  ac.     (Distributive  law  for  multiplication 
with  respect  to  addition.) 

(6)  If  X  follows  0,  then  a  -\-  x  follows  a;  and  if  x  precedes  0,  then 
a  -{-  X  precedes  a. 

(7)  If  a  precedes  b,  there  is  an  element  x  which  comes  after  0  such 
that  a  -{-  X  =  b,  and  an  element  y  which  comes  before  0  such  that 
a  =-  b  -\-y. 

(8)  If  a  and  b  both  come  after  0,  then  their  product,  ab,  also 
comes  after  0. 

34.  As  examples  of  the  use  of  mathematical  induction,  I  give 
the  proofs  of  the  first  two  theorems  in  §  33. 

Proof  of  theorem  1.  First,  if  the  theorem  is  true  for  c  =  n,  then 
it  will  be  true  for  c  =  n',  where  n'  denotes,  for  the  moment,  the 
successor  of  n. 

For,  if  we  denote  +1  simply  by  1,  we  have: 

(a  +  6)  +  n'     =  [(a  +  6)  +  n]  +  1  (by  definition) 

=  [a  +  (6  +  w)]  +  1  (by  hypothesis) 

=  a  +  [(6  +  n)  +  1]  (by  definition) 

=  a  +  [6  4-  (n  +  1)]  (by  definition) 
=  a  +  (6  +  n'). 

Secondly,  the  theorem  is  clearly  true  for  c  =  1,  by  the  definition 
of  sum.  Therefore,  by  the  first  part  of  the  proof,  since  it  is  true  for 
c  =  1,  it  will  be  true  for  c  =  2;  and  being  true  for  c  =  2,  it  will  be 
true  f or  c  =  3 ;  and  so  on.  In  this  way  the  truth  of  the  theorem  for 
any  given  value  of  c  can  be  established,  since  by  the  theorem  of 
mathematical  induction  there  is  no  element  c  which  cannot  be 
reached  in  this  manner. 

Proof  of  theorem  2.    We  establish  first  the  lemma  that  1  +  a  = 
a  +  1  by  the  same  method  of  "  proof  from  n  to  w  +  1,"  using  the 
equations 
n'  +  1  =  (n  +  1)  +  1  =  (1  +  ri)  +  1  =  1  +  (n  +  1)  =  1+  w'. 


§35  DISCRETE  SERIES  29 

The  proof  of  the  main  theorem,  that  a  -\-  b  =  b  -\-  a,  then  follows 
m  a  similar  way  from  the  equations 
a  +  n'  =  a  +  (n  +  1)  =  (a  +  n)  +  1  =  (n  +  a)  +  1 

=  n  +  (a  +  1)  =  n  +  (1  +  a)  =  (n -\- 1) -\- a  =  n'  +  a. 

The  proofs  of  the  remaining  theorems  involve  no  new  difficulty 
and  can  be  readily  supplied  by  the  reader;  when  these  eight  theo- 
rems have  once  been  established,  the  further  development  of  the 
theory  follows  lines  that  are  familiar  from  any  text-book  of  arith- 
metic and  need  not  be  repeated  here.*  The  system  (§  11)  thus  de- 
termined is  called,  with  reference  to  the  arbitrary  origin  0,  the 
algebra  of  all  integers,  with  regard  to  <,  +,  and  X. 

35.  Turning  now  to  the  progressions, f  there  are  two  principal 
methods  of  introducing  the  notions  of  sum  and  product,  leading  to 
two  different  systems  (K,  <,  +,  X).  In  both  systems  the  sums 
and  products  are  defined  absolutely,  in  terms  of  the  relation  of  order 
(see  §31). 

In  the  first  theory,  the  progression  is  denoted  by 
12    3 
the  sums  and  products  being  defined  as  follows : 

Sum:  a  -\-  1  =  the  successor  of  a;  a  -\-  2  =  the  successor  of 
a -\-  1;  and  so  on;  in  general, 

a  +  (the  successor  of  n)  =  the  successor  of  (a  +  n). 

Product:   I  X  a  =  a;  2Xa=la  +  a;  and  so  on;  in  general, 

(the  successor  of  n)  X  a  =  na  -{-  a. 
This  system  is  called  the  algebra  of  the  positive  integers,  with 
regard  to  <,  +,  and  X. 

In  the  second  theory,  the  progression  is  denoted  by 
0,  1,  2,  3,  ...  , 

the  sums  and  products  for  elements  other  than  0  being  defined  as 
above,  and  a  +  0  =  0  +  a  =  a  and  aXO  =  OXa  =  0. 

*  See  O.  Stolz  and  G.  A.  Gmeiner,  Theoretische  Arilhmetik  (1901-     ). 

t  We  pass  over  the  regressions  without  separate  discussion,  since  whatever 
is  true  of  a  progression  is  true  of  a  regression  if  the  words  "  before  "  and  "  after," 
etc.,  are  interchanged. 


30  TYPES  OF  SERIAL  ORDER  §36 

This  system  is  called  the  algebra  of  the  positive  integers  with  zero, 
with  regard  to  <,  +,  and  X. 

In  both  theories,  theorems  1-5  of  §  33  hold  without  change, 
theorems  6-7  have  to  be  slightly  modified  (in  an  obvious  way),  and 
theorem  8  is  superfluous;  the  further  development  of  the  subject 
need  not  detain  us  here. 

36.  In  view  of  §§  30-35  it  is  interesting  to  note  the  relation 
between  the  system  of  natural  numbers  (which  has  been  assumed 
as  famihar,  for  purposes  of  illustration,  throughout  the  book),  and 
the  ordinal  theory  of  progressions  (§  24).  This  relation  may  be 
stated  as  follows: 

If  the  class  of  natural  numbers  in  the  usual  order  —  from  what- 
ever source  it  may  be  derived  —  is  assumed  to  be  a  system  which 
satisfies  the  conditions  1-3,  and  Nl-NS,  and  has  a  first  element  but 
no  last,  then  it  may  be  regarded  as  the  typical  example  of  a  progres- 
sion, and  all  the  theorems  which  can  be  estabHshed  for  any  progres- 
sion will  apply  to  the  system  of  natural  numbers.  The  question 
whether  the  system  of  natural  numbers,  as  commonly  conceived, 
does  actually  possess  the  properties  demanded  in  these  eight  postu- 
lates is  a  question  for  the  psychologist  or  the  epistemologist  to 
decide;  as  far  as  the  mathematician  is  concerned,  the  theory  of  the 
natural  numbers,  in  its  abstract  form,  can  be  derived  wholly  from 
the  set  of  postulates  just  mentioned,  the  concrete,  empirical  system 
of  natural  numbers  being  used  only  as  a  means  of  establishing  the 
consistency  of  these  postulates. 

Denumerable  classes 

37.  Any  i«finite  class  the  elements  of  which  can  be  put  into  one- 
to-one  correspondence  with  the  elements  of  a  progression  (§  24)  is 
said  to  be  denumerable  (abzahlbar,  denombrahle,  enumerable,  numer- 
able, countable).* 

In  other  words,  if  we  assume  that  the  natural  numbers  in  their 
usual  order  form  a  progression  (§  36),  a  denumerable  class  is  one 

*  This  notion  was  introduced  by  Cantor;  see  CrelWs  Journ.  fiir  Math.,  vol. 
77  (1873),  p.  258,  and  Math.  Ann.,  vol.  15  (1879),  p.  4.  For  an  extension  of  the 
notion,  see  Math.  Ann.,  vol.  23  (1884),  p.  456. 


§38  DISCRETE  SERIES  31 

which  can  be  put  into  one-to-one  correspondence  with  the  class  of 
all  natural  numbers. 

Every  class  which  appears  already  ordered  in  the  form  of  a  pro- 
gression is  ipso  facto  a  denunierable  class ;  other  classes  may  have 
to  be  ingeniously  arranged  before  they  can  be  shown  to  be  de- 
numerable;  for  example,  the  class  of  all  proper  fractions  is  shown 
to  be  denumerable  by  the  device  given  in  §  19,  6.* 

Since  any  infinite  discrete  series  can  be  arranged  as  a  progres- 
sion,! it  is  obvious  that  the  term  progression  might  be  replaced  by 
regression  or  by  unlimited  discrete  series,  in  the  definition  of  a 
denumerable  class. 

38.  The  following  are  the  principal  theorems  concerning  de- 
numerable classes :t 

(1)  If  any  finite  class  is  added  to  a  denumerable  class,  the  result- 
ing class  will  still  be  denumerable. 

For,  a  progression  remains  a  progression  when  a  finite  number  of 
elements  are  added  at  the  beginning, 

(2)  A  class  composed  of  any  finite  number  of  denumerable 
classes,  or  even  a  class  composed  of  a  denumerable  infinity  of  de- 
numerable classes,  will  itself  be  a  denumerable  class. 

For,  if  Oi,  02,  as,  ...  ;  6i,  62,  63,  ...  ,  etc.,  are  the  component 
classes,  we  have  merely  to  arrange  the  elements  of  the  whole  class 
in  a  two-dimensional  array,  as  in  the  diagram, 

tti,  Oi,  as,  ... 

&1,    &2,    &3,    .    .    . 
Cl,    C2,    C3,    .    .    . 


and  then  read  the  table  diagonally  thus : 

1         2     3         4      5     6... 
ai        02    61        az    bi    Cl     .  .  .  . 

*  Cf.  G.  Faber,  Math.  Ann.,  vol.  60  (1905),  p.  196. 

t  To  arrange  an  unlimited  discrete  series  as  a  progression,  take  the  elements 
alternately.  Of  coxirse  the  correspondence  will  not  be  one  which  preserves  the 
relations  of  order. 

J  G.  Cantor,  Crelle's  Journ.  fiir  Math.,  vol.  84  (1877),  p.  243. 


32  TYPES  OF  SERIAL  ORDER  §39 

(3)  Any  collection  of  non-overlapping  three-dimensional  regions 
of  space  is  at  most  denumerably  infinite.* 

From  this  theorem  we  have  the  important  corollary  that  every 
collection  of  material  objects  is  at  most  denumerably  infinite;  hence, 
if  we  wish  to  find  an  example  of  a  non-denumerably  infinite  class, 
we  must  seek  it  among  the  classes  whose  elements  are  ideal,  not 
material,  entities. 

The  proof  of  the  theorem  is  as  follows : 

Case  I,  when  the  given  collection  C  Hes  wholly  inside  a  finite 
sphere,  with  center  at  0  and  radius  r.  —  Consider  the  denumerable 
series  of  intervals  between  the  numbers 

V,     V/2,     F/4,     V/8,     F/16,  .  .  .  , 

where  V  is  the  volume  of  the  sphere.  The  number  of  elements  of  C 
which  he  between  F/2"+^  and  F/2"  in  volume  is  at  most  finite 
(since  otherwise  the  volume  of  the  whole  collection  C  would  be 
greater  than  V) ;  therefore,  by  theorem  2,  the  number  of  elements 
in  the  whole  collection  C  is  at  most  denumerably  infinite. 

Case  II,  when  the  given  collection  C  Hes  wholly  outside  the 
sphere.  —  This  case  can  be  reduced  to  Case  I  by  an  "  inversion  " 
of  space  with  respect  to  the  sphere.  (An  "  inversion  "  transforms 
every  point  P  outside  the  sphere  into  another  point  P'  inside  the 
sphere,  such  that  P'  hes  on  the  fine  OP,  and  OP'  X  OP  =  r^;  this 
transformation  is  clearly  continuous,  so  that  points  which  form  a 
connected  region  outside  the  sphere  will  be  transformed  into  points 
which  form  a  connected  region  inside  the  sphere.) 

Case  III,  when  the  given  collection  Hes  partly  within  and  partly 
without  the  sphere.  —  Since  each  part  of  the  collection  is  at  most 
denumerably  infinite,  by  Cases  I  and  II,  the  whole  collection  will 
be  at  most  denumerably  infinite,  by  theorem  2. 

Analogous  theorems  hold  for  areas  in  a  plane,  or  for  segments  on 
a  Hne. 

39.  A  striking  example  of  a  denumerable  class  (though  it  in- 
volves more  knowledge  of  algebra  than  I  wish  to  assume  in  this 
book)  is  the  class  of  all  "  algebraic  numbers,"  that  is,  the  class  of 
aU  complex  quantities  which  can  be  roots  of  any  algebraic  equation 
with  integral  coefficients,  f 

*  Cantor,  Math.  Ann.,  vol.  20  (1882),  p.  117. 

t  G.  Cantor,  CreUe's  Journ.  fur  Math.,  vol.  77  (1873),  p.  258. 


§40  DISCRETE  SERIES  33 

For,  the  class  of  values  any  coefficient  can  take  on  is  denumer- 
able,  hence  the  class  of  different  equations  of  the  n^^  degree  is  de- 
numerable;  and  since  an  equation  of  the  n"*  degree  cannot  have 
more  than  n  roots,  the  class  of  all  the  roots  of  all  equations  of  the 
n*^  degree  is  denumerable;  and  finally  the  class  of  possible  degrees 
is  denumerable,  so  that  the  whole  class  of  all  the  roots  of  all  alge- 
braic equations  is  denumerable. 

40.  An  example  of  a  non-denumerable  class  is  the  class  of  all 
non-terminating  decimal  fractions  (see  §  19,  9) .  For,  if  we  suppose 
that  this  class  is  denumerable,  every  non-terminating  decimal 
fraction  would  have  a  definite  rank  in  a  certain  progression;  but  if 
we  represent  this  progression  as  follows: 


1. 

0.  tti  a^  as  . 

.  . 

2. 

0.  6i  62  &3  . 

.  . 

3. 

0.  Ci  C2  C3  . 

.  . 

where  each  letter  (with  subscript)  denotes  one  of  the  digits  0,  1,  2, 
.  .  .  ,  9,  we  can  at  once  describe  non-terminating  decimals  which 
do  not  belong  to  this  list.     Thus  the  decimal 

0.  a;i  a:2  rc3  .  .  .  ,  j 

^  where  Xi  is  different  from  ai,  Xi  different  from  62,  3:3  different  from 
C3,  etc.,  has  no  place  in  the  progression,  since  it  differs  from  the  n**" 
decimal  in  at  least  the  n*^  digit.* 

Therefore  the  class  of  decimals  cannot  be  denumerable. 

*  G.  Cantor,  Jahresbericht  der  D.  Maih.-Ver.,  vol.  1  (1892),  p.  75. 


raAPTER  IV 

Dense  Series:  Especially  the  Type  n  of  the 
^  Rational  Numbers 

41.  In  this  chapter  we  consider  series  {K,  < )  which  satisfy  the 
general  postulates  1-3  of  §  12,  and  also  the  special  postulates  HI 
and  H2,  below ;  the  properties  here  demanded  being  quite  different 
from  the  properties  of  the  discrete  series  considered  in  the  last 
chapter. 

Postulate  HI*  If  a  and  h  are  elements  of  the  class  K,  and 
a  <  h,  then  there  is  at  least  one  element  x  in  K  such  that  a  <  x  and 
X  <  b. 

Any  series  which  has  this  property  is  said  to  be  dense.'f  Between 
every  two  elements  of  a  dense  series  there  wiU  be  at  least  one  and 
therefore  an  infinity  of  other  elements;  so  that  no  element  has  a 
successor,  and  no  element  a  predecessor. 

Postulate  H2.  The  class  K  is  denumerable;  that  is,  the  ele- 
ments of  K  can  be  put  into  one-to-one  correspondence  with  the 
elements  of  a  progression  (§  37) . 

Any  series  which  satisfies  these  two  postulates  HI  and  H2  is 
called  a  denumerable  dense  series,  or  more  briefly,  a  rational  series. 

A  series  whose  elements  form  a  denumerable  class  may  be  called, 
for  brevity,  a  denumerable  series. 

42.  The  simplest  example  of  a  series  which  is  both  denumerable 
and  dense  is  the  class  of  proper  fractions  arranged  in  the  usual  order 
(see  §  19,  5).    For,  if  a  =  m/n  and  b  =  p/q,  and  a  <  b,  then  there 

are  elements  x  which  he  between  a  and  b  (for  example,  x  =  — ; — , 

n  -\-  q 

*  The  letter  H  is  intended  to  suggest  the  type  17  (§  44). 
t  Cantor's  term  is  uberall  dicht.    Weber  uses  dicht,  which  Russell  replaces 
by  compact;  Principles  of  Mathematics,  vol.  1,  p.  271.    See  however,  §  62a. 

34 


§45  DENSE  SERIES  35 

reduced  to  its  lowest  terms) ;  and  on  the  other  hand,  if  we  arrange 
the  elements  in  a  two-dimensional  array,  and  then  read  the  table 
diagonally,  as  in  §  38,  we  see  at  once  that  the  class  is  denumer- 
able.*     (Compare  §  19,  6.) 

11111 

2  3  4  5  6  *  *  * 

2  2  2  2^ 

3  5  7  9        11  ■  *  * 

3  3  3  3^ 

4  5  7  8        10  '  *  * 


43.  In  every  series  of  this  sort  we  have  to  do,  strictly  speaking, 
with  two  serial  relations:  with  respect  to  one,  the  series  is  dense; 
with  respect  to  the  other,  the  series  is  a  progression. 

44.  The  type  rj.  All  denumerable  dense  series,  like  all  discrete 
series,  can  be  divided  into  four  groups,  distinguished  by  the  pres- 
ence or  absence  of  first  and  last  elements.  All  the  series  of  any  one 
of  these  four  groups  are  ordinally  similar,  as  we  shall  prove  below, 
and  therefore  constitute  a  definite  type  of  order.  In  particular,  the 
type  of  denumerable  dense  series  with  neither  extreme  is  called  by 
Cantor  the  type  t]. 

The  simplest  example  of  a  series  of  the  type  rj  is  the  class  of 
proper  fractions  in  the  usual  order  as  already  mentioned.  By 
adding  an  element  0/1  at  the  beginning,  or  an  element  1/1  at  the 
end,  or  both,  we  have  an  example  of  a  denumerable  dense  series 
with  a  first  element,  or  a  last  element,  or  both.  Other  examples 
will  be  given  in  §  51. 

45.  We  now  give  the  proof  f  that  any  two  denumerable  dense 
series  are  ordinally  similar,  provided  they  agree  in  regard  to  the 
presence  or  absence  of  extreme  elements;  it  will  clearly  be  sufficient 
to  consider  two  series  of  the  type  77,  having  neither  extreme. 

*  Cantor,  Crelle's  Journ.  fiir  Math.,  vol.  84  (1877),  p.  250. 
t  Cantor,  Math.  Ann.,  vol.  46  (1895),  §  9,  p.  504. 


36  TYPES  OF  SERIAL  ORDER  §45 

Let  the  two  given  series  be  A  and  B;  and  let  the  terms  of  each, 
when  rearranged  in  the  form  of  a  progression,  be  denoted  by 

Oi,    02,    fls,    .    .    , 

and 

&1,     &2,     &3,      .     .     .     . 

In  order  to  estabhsh  a  one-to-one  correspondence  between  A  and 
5  in  a  manner  preserving  order,  we  proceed  step  by  step,  as  follows, 
it  being  understood  that  any  step  is  to  be  omitted  if  the  element 
considered  has  already  been  assigned : 

To  ai  assign  the  element  bi,  and  to  6i  assign  the  element  au 

The  elements  ai  and  6i  then  divide  each  of  the  original  series  A 
and  B  into  tv/o  sections. 

As  to  02,  we  find  in  which  of  the  two  sections  of  A  it  belongs,  and 
assign  to  it  the  first  of  the  unused  6's  which  belongs  in  the  corre- 
sponding section  of  B;  and  as  to  62  (if  not  already  assigned),  we 
find  in  which  section  of  B  it  belongs,  and  assign  to  it  the  first  of  the 
unused  a's  which  belongs  in  the  corresponding  section  of  A. 

The  elements  tti  and  02  then  divide  the  series  A  into  three  sections 
(1st,  2d,  and  3d),  while  the  elements  61  and  62  divide  the  series  B 
into  three  corresponding  sections  (1st,  2d,  and  3d).  As  to  as,  if  not 
already  assigned,  we  find  in  which  of  the  three  sections  of  A  it  be- 
longs, and  assign  to  it  the  first  of  the  (unused)  6's  which  belongs  in 
the  corresponding  section  of  B.  Then  as  to  63,  if  not  already  as- 
signed, we  find  in  which  of  the  three  sections  of  B  it  belongs,  and 
assign  to  it  the  first  of  the  (unused)  a's  which  belongs  in  the  corre- 
sponding section  of  A. 

And  so  on.  After  2n  steps,  the  first  n  of  the  a's  will  have  been 
assigned  and  will  divide  A  into  n  -\-  1  sections,  and  the  first  n  of  the 
fe's  will  have  been  assigned  and  will  divide  B  into  n  +  1  corre- 
sponding sections.  Then  as  to  a„+i,  if  not  already  assigned,  we 
find  in  which  of  the  n  -\-  1  sections  of  A  it  belongs,  and  assign  to  it 
the  first  of  the  (unused)  6's  which  belongs  in  the  corresponding 
section  of  B.  And  as  to  6„+i,  if  not  already  assigned,  we  find  in 
which  of  the  n  +  1  sections  of  B  it  belongs,  and  assign  to  it  the  first 
of  the  (unused)  a's  which  belongs  to  the  corresponding  section  of 
A. 

The  elements  called  for  at  each  stage  of  this  process  will  always 
exist,  since  in  any  series  of  type  17  there  are  elements  before  and 
after  any  given  element,  and  between  any  two  given  elements;  and 
by  the  theorem  of  mathematical  induction  as  applied  to  progres- 
sions no  element  of  either  class  is  left  out  in  the  assignment. 


§48  DENSE  SERIES  37 

It  should  be  noticed  that  the  correspondence  between  two  series 
of  type  77  can  be  set  up  in  an  infinite  number  of  ways  (compare  the 
case  of  the  unlimited  discrete  series,  §  26). 

Segments  of  series 

46.  In  the  following  sections  we  define  a  few  technical  terms 
which  will  be  of  great  service  in  the  study  of  dense  and  continuous 
series. 

In  any  series  (§  12)  a  part  C  (§  6)  which  has  the  following  prop- 
erties we  shall  call  a /wndamentoZ  segment  of  the  series:  (1)  C  is  such 
that  if  X  is  any  element  belonging  to  C,  then  every  element  that 
precedes  x  also  belongs  to  C;  and  (2)  C  has  no  last  element. 

Roughly  speaking,  a  fundamental  segment  is  a  part  of  the  series 
beginning  at  the  beginning,  and  taking  in  everything  as  far  as  it 
goes,  but  having  no  last  element.* 

47.  A  segment  in  general  may  be  defined  as  any  part  (7  of  a  series 
having  the  following  property:  if  a  and  b  are  any  two  elements 
belonging  to  C,  then  every  element  that  hes  between  a  and  b  also 
belongs  to  C. 

A  segment  C  such  that  if  a  belongs  to  C,  then  every  element  that 
f  precedes  1  ^  ^^^^  ^^j  ^^  ^   .^  ^^^^^  (  a  lower   segment )    ^ 

{ follows    J  &  >  1^  a^Q  upper  segment  J 

the  series. t 

A  fundamental  segment,  then,  is  a  lower  segment  which  has  no 
last  element. 

48.  It  will  be  noticed  at  once  that  in  some  series  no  fundamental 
segments  are  possible.  For  example,  in  a  discrete  series  (§  21)  no 
fundamental  segments  are  possible,  since  every  subclass  which 
satisfies  condition  1  of  §  46  either  has  a  last  element  or  includes  the 
whole  series.  In  other  cases  the  number  of  fundamental  seg- 
ments may  be  finite.     For  example,  in  a  series  like  this : 

*  Russell's  term  is  segment  (without  distinctive  adjective).  The  notion 
itself,  which  is  a  modification  of  Dedekind's  notion  of  a  cut  (1872),  was  intro- 
duced by  M.  Pasch  (Differential-  und  Integralrechnung,  1882),  under  the  name 
of  Zahlenstrecke.  The  term  segment  was  used  by  Peano  in  the  Formulaire  for 
1899,  p.  91,  but  seems  to  have  been  abandoned  in  later  editions. 

t  Russell,  loc.  cit.,  p.  271. 


38  TYPES  OF  SERIAL  ORDER  §49 

li,  2i,  3i,  .  .  .;  I2,  22,  82,  •  .  .;  I3,  23,  83,  .  .  .;  I4,  24; 

only  three  fundamental  segments  are  possible. 

In  a  dense  series,  however,  the  class  of  fundamental  segments  is 
always  infinite. 

49.  In  connection  with  fundamental  segments  the  following 
definition  is  important :  In  any  series,  if  there  is  an  element  x  such 
that  a  given  fundamental  segment  coincides  with  the  part  of  the 
series  which  precedes  x,  then  x  is  called  the  limit  of  the  segment. 

If  no  such  element  x  exists,  then  the  segment  has  no  limit  in  the 
given  series. 

We  may  then  distinguish  two  kinds  of  fundamental  segments: 
first,  those  that  have  a  limit  in  the  given  series;  and  secondly, 
those  that  have  not. 

50.  The  importance  of  this  distinction  between  the  two  kinds  of 
fundamental  segments  will  be  clearer  after  the  continuous  series 
have  been  discussed,  in  the  next  chapter.  For  the  present,  the  most 
important  thing  is  to  see  clearly  that  in  some  series  fundamental 
segments  of  the  second  kind  actually  exist.  To  illustrate  this  point, 
consider  the  class  of  proper  fractions  arranged  in  the  usual  order 
and  take  as  the  subclass  C  the  class  of  all  the  fractions  m/n  for 
which  2m2  is  less  than  n^ ;  this  subclass  C  will  then  be  a  fundamental 
segment  having  no  limit  in  the  given  series.* 

To  prove  this  statement,!  notice  first  that  C  satisfies  the  defini- 
tion of  a  fundamental  segment. 

For:  (1)  if  m/n  belongs  to  C,  and  p/q  precedes  m/n,  then  p/q 
also  belongs  to  C,  as  a  brief  computation  will  show;  (2)  if  m/n 
belongs  to  C,  then  there  are  fractions,  —  for  example, 

(Gm^  +  l)/6mn,  X 

*  In  the  series  of  all  real  numbers,  which  is  not  under  consideration  at  this 
point,  the  subclass  C  would  be  described  as  the  class  of  aU  the  rational  numbers 
that  precede  Vl/2.  In  verifying  the  nimierical  example  below,  note  that 
since  m  and  n  are  integers,  2to'^  must  be  less  than  n^  by  at  least  one;  that  is, 
2m^  <n^  -  1. 

t  R.  Dedekind,  Stetigkeit  und  irrationale  Zahlen,  1872;  H.  Weber,  Algebra, 
vol.  1,  p.  6. 

J  Reduced  to  its  lowest  terms. 


§51  DENSE  SERIES  39 

—  which  follow  min  and  still  belong  to  C,  so  that  C  has  no  last 
element;  and  (3)  C  is  neither  empty  nor  contains  the  whole  class, 
since  it  contains  1/4  and  does  not  contain  3/4. 

Furthermore,  there  is  no  element  xly  which  can  serve  as  the  limit 
of  the  segment.  For,  first,  if  2x^  were  less  than  if,  there  would  be 
elements  of  C,  —  for  example  (6a;2  +  \)l<Q,xy*  —  which  came  after 
xly;  secondly,  if  2x^  were  greater  than  y^,  there  would  be  elements 
ojf  the  series,  —  for  example  {^x^  —  l)/6a;?/,*  —  which  preceded 
xly  and  yet  did  not  belong  to  C;  and  thirdly,  if  2x^  =  y^,  we  should 
have  an  equation  containing  the  factor  2  an  odd  number  of  times 
on  the  left  hand  side  and  an  even  number  of  times  (if  at  all)  on  the 
right  hand  side,  which  is  impossible  in  view  of  the  fact  that  a 
natural  number  can  be  resolved  into  prime  factors  in  only  one  way. 

Hence  the  class  C  is  a  fundamental  segment  which  has  no  limit.f 

From  this  discussion  it  is  clear  that  Dedekind's  postulate  (§  21) 
is  false  in  every  series  of  type  ??;  for  (by  §  45)  any  series  of  type  17 
may  be  replaced  by  the  series  of  proper  fractions  in  the  usual  order, 
and  if  we  divide  this  series  into  two  parts,  Ki  and  K2,  so  that  Ki 
contains  every  fraction  mIn  for  which  2m?  <  'n?,  and  K2  all  the 
other  fractions,  then  there  will  be  no  element  in  the  series  which 
could  serve  as  the  element  X  required  in  Dedekind's  postulate. 

Examples  of  denumerable  dense  series 

51.  In  this  section  we  give  a  number  of  examples  of  denumerable 
dense  series;  any  one  of  these  systems  is  sufficient  to  show  the 
consistency  of  the  postulates  1-3,  H1-H2  (compare  §  19). 

In  every  denumerable  dense  series  all  the  postulates  iVl-iV3  for 
discrete  series  (§  21)  are  false  (compare  §  50). 

(1)  The  simplest  example  of  a  series  of  type  rj  is  the  class  of 
proper  fractions  in  the  usual  order,  as  already  mentioned  in  §  44. 

Other  examples  are : 

(2)  The'  class  of  (absolute)  rational  numbers  and 

(3)  the  class  of  all  rational  numbers  (positive,  negative  or  zero), 
—  both  being  arranged  in  the  usual  order. 

*  Reduced  to  its  lowest  terms. 

t  A  simpler  example  of  the  same  sort  is  provided  by  the  red  elements  in 
example  1,  §  29. 


40  TYPES  OF  SERIAL  ORDER  §51 

By  an  absolute  rational  number  we  mean  an  ordered  pair  of 
natural  numbers,  7n/7i,  in  which  the  first  number,  m,  called  the 
numerator,  and  the  second  number,  n,  called  the  denominator,  are 
relatively  prime.  By  the  usual  order  in  this  class  we  mean  that 
m/n  is  to  precede  p/q  when  m  X  qis  less  than  n  X  p. 

The  class  of  all  rationals  is  then  composed  of  three  kinds  of  ele- 
ments: (1)  the  positive  rationals,  which  are  absolute  rationals 
affected  with  the  sign  + ;  (2)  the  negative  rationals,  which  are 
absolute  rationals  affected  with  the  sign  —  ;  and  (3)  an  extra  ele- 
ment called  zero.  The  "  usual  order  "  in  this  class  is  precisely 
defined  as  follows :  of  two  positive  rationals,  that  one  shall  precede 
whose  absolute  value  would  precede  in  the  order  of  absolute  ra- 
tionals; of  two  negative  rationals,  that  one  shall  precede  whose 
absolute  value  would  follow  in  the  order  of  absolute  rationals;  of 
two  rationals  having  opposite  signs,  the  negative  precedes  the 
positive;  and  the  rational  0  follows  every  negative  rational  and 
precedes  every  positive  rational. 

The  rationals  between  0  and  1/1,  or  the  absolute  rationals  which 
precede  1/1,  are  the  proper  fractions  (§  19,  5). 

If  we  assign  to  each  absolute  rational  number  p/q  the  proper 
fraction  p/(p  +  g),  we  thereby  establish  an  ordinal  correspondence 
between  the  series  of  absolute  rationals  and  the  series  of  proper 
fractions,  in  accordance  with  the  theorem  of  §  45.  This  done,  an  L^Y-a 
ordinal  correspondence  between  the  series  of  absolute  rationals  and 
the  series  of  all  rationals  can  be  readily  established. 

(4)  As  another  example  of  a  series  of  type  77,  consider  the  class  of 
points  lying  within  a  one-inch  square,  and  such  that  their  distances, 
X  and  y,  from  two  sides  of  the  square  are  proper  fractions  of  an  inch; 
and  let  the  points  be  arranged  in  order  of  magnitude  of  the  x's,  or 
in  case  of  equal  a;'s,  in  order  of  magnitude  of  the  y^s. 

This  system  clearly  satisfies  all  the  postulates  for  a  series  of  type 
7] ;  it  ought  therefore  to  be  possible  to  exhibit  an  ordinal  correspond- 
ence between  this  system  and  the  series  of  proper  fractions. 

This  may  be  done  as  follows.*  Starting  with  a  line  AB  of  fixed 
length,  mark  the  middle  third  of  it ;  then  mark  the  middle  third  of 
each  of  the  two  remaining  parts,  then  the  middle  third  of  each  of 

*  Compare  §  52,  3,  below.  The  device  is  due  to  H.  J.  S.  Smith,  Proc.  Lond. 
Math.  Soc,  vol.  6  (1875),  p.  147;  cf.  G.  Cantor,  Math.  Ann.,  vol.  21  (1883), 
p.  590,  note  11,  and  W.  H.  Young,  Proc.  Lond.  Math.  Soc,  vol.  34  (1902),  p.  286. 


U 


§52  DENSE  SERIES  41 

the  four  remaining  parts ;  and  so  on.  The  class  of  marked  sections 
of  the  hne  is  then  a  deniimerable  class,  which  forms  a  dense  series 
of  type  I?  along  the  line  AB.  Now  the  vertical  lines  in  the  given 
square,  corresponding  to  fractional  values  of  x,  also  form  a  de- 
numerable  series  of  type  rj;  hence,  by  §45,  the  class  of  vertical 
lines  can  be  brought  immediately  into  ordinal  correspondence 
with  the  class  of  marked  sections  of  the  line  AB.  It  remains  merely 
to  determine  on  each  section  the  class  of  what  we  may  call,  for  the 
moment,  its  "fractional"  points,  that  is,  the  class  of  points  whose 


distances  from  one  end  of  the  section  are  fractional  parts  of  the 
length  of  th6  section;  this  class  of  points  can  then  be  brought 
into  ordinal  correspondence  with  the  ''fractional"  points  of  the 
corresponding  vertical  line  in  the  square  by  a  suitable  magnifica- 
tion. 

The  given  series  of  points  in  the  square  is  thus  reduced  to  a  dense 
series  of  points  on  the  line  AB. 

By  a  double  application  of  the  same  method,  the  "  frac- 
tional "  points  within  a  cube  can  be  treated  in  a  similar  way. 

Examples  of  series  which  are  not  denumerahle  and  dense 

52.  The  following  examples  of  series  which  fail  to  satisfy  one  or 
both  of  the  postulates  HI  and  H2  show  that  these  postulates  are 
independent  of  each  other  (compare  §  20). 

(1)  Denumerahle  series  which  are  not  dense. 

(a)  One  example  of  this  kind  is  any  unlimited  discrete  series, 

^"*  ^'         ....  -3.  -2,  -1,    0,  -1,  -2,  -3,  .  .  .  . 

By  adding  an  element  ~z  at  the  beginning,  or  an  element  +z  at 
the  end,  or  both,  we  obtain  an  example  with  a  first  or  a  last  ele- 
ment, -y  -'oth.     Progressions  and  regressions  are  also  examples. 

(6)  Another  example  is  a  class  composed  of  two  sets  of  proper 
fractions,  say  red  and  blue,  with  the  relation  of  order  defined  as 
follows :  of  two  elements  which  have  unequal  absolute  values  that 
one  shall  precede  which  would  precede  in  the  usual  order  of  proper 


42  TYPES  OF  SERIAL  ORDER  §52 

fractions,  regardless  of  color;  of  two  elements  which  have  the  same 
absolute  value,  the  red  shall  precede. 

This  system  is  built  up  by  interpolating  the  elements  of  one 
dense  series  between  the  elements  of  another  dense  series;  the  re- 
sulting series,  instead  of  being  "  more  dense,"  as  one  might  have 
been  tempted  to  expect,  has  lost  the  property  of  density  altogether, 
since  every  red  element  has  an  immediate  successor. 

(2)  Dense  series  which  are  not  denumerahle. 

(a)  The  class  of  non-terminating  decimal  fractions  arranged  in 
the  usual  order  (see  §  19,  9)  is  a  dense  series,  which  we  have  already 
shown  to  be  non-d enumerable  (§  40). 

(6)  Another  example  is  obtained  from  example  (3),  below,  by 
omitting  the  "  points  of  division  "  that  form  a  part  of  that  class. 

(c)   For  another  example,  see  §  64,  3,  (6),  footnote. 

(3)  A  series  which  is  neither  denumerahle  nor  dense. 

A  striking  example  of  a  series  which  is  neither  denumerahle  nor 
dense  may  be  constructed  as  follows:  *  Starting  with  a  line  one 
inch  long,  mark  the  middle  third  of  it;  then  mark  the  middle  third 
of  each  of  the  two  remaining  parts,  then  the  middle  third  of  each  of 
the  four  remaining  parts,  and  so  on  (§  51,  4);  the  class  considered 
contains  (1)  all  the  points  of  division,  and  (2)  all  the  unmarked 
points  of  the  line ;  and  the  order  of  the  points  is  the  natural  order 
along  the  line. 

This  series  is  clearly  not  dense,  since  if  a  and  h  are  the  end-points 
of  one  of  the  marked  sections,  there  is  no  point  of  the  series  which 
lies  between  them;  indeed,  no  segment  of  the  series  will  be  dense, 
since  every  segment  (§  47)  will  contain  a  marked  section  of  the  line. 
On  the  other  hand,  the  class  is  not  denumerahle;  the  proof  of  this 
fact  (which  requires  a  little  more  mathematics  than  is  properly 
assumed  in  this  book)  may  be  outlined  as  follows: 

Let  the  distance  from  one  end  of  the  line  to  each  point  of  the  line 
be  represented  by  a  ternary  fraction  (instead  of  a  decimal  fraction) 
of  an  inch;  that  is,  by  a  (finite  or  an  infinite)  expression  of  the  form 

0.  ai  02  03  .  .  .  , 
*  Cf.  footnote  under  §  51,  4. 


§53  DENSE  SERIES  43 

in  which  ai  shows  the  number  of  thirds,  a2  the  number  of  ninths, 
as  the  number  of  twenty-sevenths,  and  in  general  a„  the  number  of 
(l/3")ths;  the  digits  ai,  a^,  as,  etc.,  being  allowed  to  take  any  of  the 
three  values  0,  1,  and  2.  It  can  then  be  shown,  by  a  computation 
involving  only  an  elementary  knowledge  of  the  so-called  geometric 
series,  that  the  points  of  the  marked  sections  of  the  line  (without 
the  points  of  division)  correspond  to  precisely  those  ternary  frac- 
tions in  which  the  digit  1  occurs;  the  points  of  our  class,  therefore, 
correspond  to  the  ternary  fractions  in  which  the  digits  0  and  2  only 
are  used;  and  this  class  can  be  shown  to  be  non-denumerable  by 
the  method  employed  in  §  40  for  the  decimal  fractions. 

Arithmetical  operations  among  the  elements  of  a  dense  series 

53.  In  conclusion,  we  notice  that  since  the  theorem  of  mathe- 
matical induction  does  not  apply  to  dense  series,  it  is  not  possible 
to  give  purely  ordinal  definitions  for  the  sums  and  products  of  the 
elements  of  such  a  series.  All  that  we  could  do  in  this  direction 
would  be  to  define  the  sums  and  products  of  the  elements  of  some 
particular  dense  series,  say  the  series  of  the  rational  numbers  in  the 
usual  order,  by  the  use  of  some  extra-ordinal  properties  peculiar  to 
that  series;  then  since  all  series  of  type  t;  are  ordinally  similar,  the 
definitions  set  up  in  the  standard  series  could  be  transferred  to  any 
other  series  of  the  same  type  by  a  one-to-one  correspondence.  This 
method  would  be  wholly  inadequate,  however,  since  the  ordinal 
correspondence  could  be  set  up  in  an  infinite  number  of  ways. 
Indeed,  in  the  case  of  a  series  of  type  r/  (without  extreme  elements), 
unless  we  introduce  some  other  fundamental  notion  beside  the 
notion  of  order,  the  elements  have  no  ordinal  properties  by  which 
we  can  tell  them  apart.  It  is  better,  therefore,  to  introduce  addition 
and  multiplication  as  fundamental  notions  of  the  system  (compare 
§  11),  and  define  their  properties  by  postulates;  this  problem  is, 
however,  beyond  the  scope  of  the  present  work.* 

*  See,  for  example,  my  two  monographs  cited  in  the  introduction. 


CHAPTER  V 

Continuous  Series:  Especially  the  Type  6  of  the 
Real  Numbers 

54.  In  the  preceding  chapters  we  have  considered  the  discrete 
series  (§21)  and  the  dense  series  (§  41) ;  we  turn  now  to  the  study 
of  the  hnear  continuous  series,  which  are  the  most  important  for 
algebra. 

A  continuous  series  in  general  is  defined  as  any  series  which  satis- 
fies postulates  1-3  of  §  12,  and  also  Dedekind's  postulate  (CI, 
below)  and  the  postulate  of  density  (C2) ;  a  linear  continuous  series 
is  then  any  continuous  series  which  satisfies  also  a  further  condition, 
which  I  shall  call  the  postulate  of  linearity  (C3). 

Postulate  CI.*  (Dedekind's  postulate.)  If  Ki  and  K2  are  any 
tivo  non-empty  parts  of  K,  such  that  every  element  of  K  belongs 
either  to  Kt  or  to  K2  and  every  element  of  Ki  precedes  every  element  of 
K2,  then  there  is  at  least  one  element  X  in  K  such  that: 

(1)  any  element  that  precedes  X  belongs  to  Ki,  and 

(2)  any  element  that  follows  X  belongs  to  K^. 
This  is  the  same  as  postulate  Nl  in  §  21. 

Postulate  C2.  {Postulate  of  density.)  If  a  and  b  are  elements  of 
the  class  K,  and  a  <  b,  then  there  is  at  least  one  element  x  in  K  such 
that  a  <  X  and  x  <  b. 

This  is  the  same  as  postulate  /Z^l  in  §  41. 

Postulate  CS.f  (Postulate  of  linearity.)  The  class  K  contains 
a  denumerable  subclass  R  (§  37)  in  such  a  way  that  between  any  two 
elements  of  the  given  class  K  there  is  an  element  of  R. 

*  R.  Dedekind,  loc.  cit.  (1872). 

t  G.  Cantor,  loc.  cit.  (1895),  §  11,  p.  511.  O.  Veblen  replaces  this  postulate 
of  linearity  by  two  other  postulates  which  he  calls  the  pseudo-Archimedean 
postulate  and  the  postulate  of  uniformity  [Trans.  Amer.  Math.  Soc,  vol.  6 
(1905),  pp.  165-171].  See  also  R.  E.  Root,  Limits  in  terms  of  order,  Trans. 
Amer.  Math.  Sac,  vol.  15  (1914),  pp.  51-71. 

44 


§56 


CONTINUOUS  SERIES 


45 


The  consistency  and  independence  of  these  postulates  will  be 
discussed  in  §  63  and  §  64;  postulate  C2  is  clearly  redundant  when- 
ever postulate  C3  is  assumed. 

55.  The  most  familiar  example  of  a  linear  continuous  series  is  the 
class  of  points  on  a  line,  say  one  inch  long,  the  relation  a  <  h  sig- 
nifying that  a  lies  on  the  left  of  6.  Dedekind's  postulate  is  satis- 
fied in  this  system,  since  if  Ki  and  K2  are  two  parts  of  the  kind 
described  in  the  postulate,  there  will  be  a  point  of  division  on  the 
line  (either  the  last  point  of  Ki  or  the  first  point  of  Ko),  which  will 
serve  as  the  point  X  demanded  in  the  postulate.  The  postulate  of 
density  is  also  clearly  satisfied,  since  between  any  two  points  of  the 
line  other  points  can  be  found.  Finally,  to  see  that  the  postulate 
of  linearity  holds,  take  as  the  subclass  R  the  class  of  all  points 
of  the  line  whose  distances  from  one  end  are  rational  fractions  of 
an  inch. 

An  example  of  a  continuous  series  which  is  not  linear  is  the  class 
of  all  points  (x,  y)  of  a  square  (including  the  boundaries),  arranged 


in  order  of  magnitude  of  the  a;'s,  or,  in  case  of  equal  x'^,  in  order 
of  magnitude  of  the  ^'s.  This  series  is  continuous  (satisfying  pos- 
tulates CI  and  C2),  but  no  subclass  R  of  the  kind  demanded  in 
postulate  C3  is  possible  within  it;  for,  if  there  were  such  a  subclass 
it  would  have  to  contain  elements  corresponding  to  every  point  of 
the  base  of  the  square  and  therefore  could  not  be  denumerable  (see 
§  58  below). •'^ 

Other  examples,  not  depending  on  geometric  intuition,  will  be 
given  in  §  63  and  §  64,  3. 

56.  With  the  aid  of  the  following  definition,  we  may  state  two 
theorems  that  hold  for  all  continuous  series. 


i .  /f  ^i 


46  TYPES  OF  SERIAL  ORDER  §56 

Definition.  Let  C  be  any  non-empty  subclass  in  any  series 
(K,  <);    if  there  is  an  element  X  in  the  series  such  that : 

(1)  there  is  no  element  of  C  which  follows  X,  while 

(2)  if  Y  is  any  element  preceding  X  there  is  at  least  one  element 
of  C  which  follows  Y:  —  then  this  element  X  is  called  the  upper 
limit  of  the  subclass  C. 

If  the  subclass  C  happens  to  have  a  last  element,  this  element 
itself  will  be  the  upper  limit  of  the  subclass.  If  C  has  no  last  ele- 
ment, it  may  or  may  not  have  an  upper  limit;  if  it  has  an  upper 
limit,  then  this  upper  limit  is  the  element  which  comes  next  after 
the  subclass  C  in  the  given  series.* 

Theorem  1.  In  any  continuous  series,  if  C  is  any  subclass  all  of 
whose  elements  precede  a  given  element,  then  C  will  have  an  upper 
limit  in  the  series. 

Briefly,  this  theorem  tells  us  that  in  any  continuous  series,  every 
subclass  which  has  any  upper  bound  will  have  a  lowest  upper 
bound,  —  the  terms  "  upper  limit  "  and  "  lowest  upper  bound  " 
being  synonymous. 

The  fuU  meaning  of  this  theorem  will  be  clearer  after  a  study  of 
the  examples  given  in  §§  63-64  of  series  that  are  and  those  that  are 
not  continuous  (compare  also  §  50) ;  the  formal  proof  is  easily 
given,  as  follows : 

Under  the  conditions  stated,  the  given  series  can  be  divided  into 
two  non-empty  subclasses,  Ki  and  K^,  the  first  containing  every 
element  that  is  equaled  or  surpassed  by  any  element  of  C,  and  the 
second  containing  all  the  other  elements;  f  then  by  Dedekind's 
postulate  there  must  be  at  least  one  element  X  "  dividing  "  Ki 
from  K2',  moreover,  there  cannot  be  two  such  elements,  for  if  there 
were,  one  would  be  the  last  element  of  Ki  and  the  other  the  first 
element  of  K2,  so  that  no  element  would  lie  between  them  (contrary 
to  the  postulate  of  density).  This  dividing  element  X  is  then  the 
element  required  in  the  theorem. 

*  It  should  be  noticed  that  this  definition  of  a  limit  of  a  subclass  in  general 
is  consistent  with  the  definition  already  given  for  the  limit  of  a  fundamental 
segment  (§  49). 

t  The  subclass  K2  will  not  be  an  empty  class,  since  by  hypothesis  there  is  at 
least  one  element  in  K  which  follows  all  the  elements  of  C. 


§58  CONTINUOUS  SERIES  47 

Similarly,  we  may  define  the  lower  limit  of  a  subclass,  and  prove 
the  analogous  theorem: 

Theorem  2.  In  any  continuous  series,  if  C  is  any  subclass  all  of 
whose  elements  follow  a  given  element,  then  C  will  have  a  lower  limit  in 
the  series. 

That  is,  in  any  continuous  series,  every  subclass  which  has  any 
lower  bound  will  have  a  highest  lower  bound,  or  lower  limit. 

Corollary.  In  any  continuous  series  which  has  a  first  and  a  last 
element,  every  subclass  will  have  both  an  upper  limit  and  a  lower  limit 
in  the  series. 

57.  The  following  theorem  gives  us  another  form  of  the  defini- 
tion of  continuous  series. 

Theorem.*  In  the  definition  of  a  continuous  series  (§  54),  Dede- 
kind^s  postulate  may  be  replaced  by  the  demand  that  every  fundamental 
segment  shall  have  a  limit  (§  49). 

For,  if  the  elements  of  the  whole  series  are  divided  into  two  sub- 
classes Ki  and  K2  as  in  the  hypothesis  of  Dedekind's  postulate,  then 
Ki  (or  Ki  without  its  last  element,  if  it  happens  to  have  one)  will  be 
a  fundamental  segment,  and  the  limit  of  this  segment  will  corre- 
spond to  the  element  X  in  Dedekind's  postulate. 

58.  The  next  theorem  concerns  the  infinitude  of  the  elements  of 
a  continuous  series. 

Theorem.  The  elements  of  any  continuous  series  (§  54)  form  an 
infinite  class  which  is  not  denumerable  (§  37) . 

The  proof,  which  is  due  to  Cantor, f  is  as  follows: 

Suppose  a  given  continuous  series  to  be  denumerable;  then  with- 
out disturbing  the  order  of  the  elements  we  may  attach  to  each  one 
a  definite  natural  number,  using  the  notation  a(n)  to  represent  the 
element  corresponding  to  the  number  n. 

We  may  assume  without  loss  of  generality  that  the  elements  have 
been  so  numbered  that  the  element  a(l)  precedes  the  element  a (2). 

Then  let  pi  and  qi  be  the  smallest  numbers  for  which  a(pi)  and 
a(qi)  lie  between  a(l)  and  a(2),  and  assume  that  the  elements  have 
been  so  numbered  that  a(pi)  <  a{qi) ;  then 

a{\)  <  a{pi)      <     a{q,)  <  a(2). 

*  Cf .  a  remark  due  to  WMtehead  in  Russell's  Principles  of  Mathematics,  vol. 
1  (1903),  p.  299,  footnote. 

t  G.  Cantor,  Crelle's  Journ.  fur  Math.,  vol.  77  (1874),  p.  260. 


48  TYPES  OF  SERIAL  ORDER  §59 

Next,  let  p2  and  92  be  the  smallest  numbers  for  which  a(p2)  and 
0(52)  lie  between  a(pi)  and  a(qi)  and  assume  a (7)2)  <  a{q2),  so  that 

a(l)  <  a(pi)  <  a(p2)      <     a(q2)  <  a{qi)  <  a(2). 

And  so  on.  In  general,  let  pk+i  and  qk+i  be  the  smallest  numbers 
for  which  a(pk+i)  and  a(qk+i)  lie  between  a{pk)  and  a(qk),  and 
assume  a(?}fc+i)<  0(5^+1).  In  this  way  we  determine  a  progres- 
sion of  elements  a(pk)  and  a  regression  of  elements  a(qk),  such 
that 

a(l)  <  a{p,)  <  a{p2)  <...<...<  aiq^)  <  a(qi)  <  a(2). 

Now  since  the  series  is  continuous,  the  progression  in  question 
ought  to  have  an  upper  limit  (§  56);  but  there  is  no  element  a(n) 
which  can  serve  as  this  upper  limit,  for  if  any  element  a(n)  is  pro- 
posed, we  can  clearly  carry  the  process  just  indicated  so  far  that 
a(w)  will  lie  outside  the  interval  a(pA;) o,{qk)'" 

Therefore  if  the  series  is  denumerable  it  cannot  be  continuous, 
and  the  theorem  is  proved. 

59.  The  theorems  of  §§  56-58  hold  for  all  continuous  series;  the 
following  'theorems  apply  only  to  the  linear  continuous  series. 

Theorem.  Every  linear  continuous  series  (§  54)  contains  a  sub- 
class R  of  type  7]  (§  44),  such  that  betiveen  any  two  elements  of  the  given 
series  there  is  an  element  of  R. 

For,  the  denumerable  subclass  R  whose  existence  is  demanded  in 
postulate  C3,  or  the  same  subclass  without  its  extreme  elements  if  it 
has  them,  is  clearly  of  type  77  (the  type  of  the  rational  numbers). 

This  subclass  R  of  type  rj  may  be  called  the  skeleton,  ov  framework, 
of  the  given  series;  the  elements  which  belong  to  R  may  be  called, 
for  the  moment,  the  rational  elements,  and  those  that  do  not  belong 
to  R  the  irrational  elements  of  the  series. 

Since  the  class  of  all  the  elements  of  any  continuous  series  is  non- 
denumerably  infinite  (§  58),  it  is  clear  that  the  rational  elements  of 
a  linear  continuous  series  cannot  exhaust  the  series;  in  fact  the 
class  of  irrational  elements  in  any  such  series  will  itself  be  non- 
denumerably  infinite  (compare  §  38). 

60.  The  most  important  property  of  the  rational  elements  is 
given  in  the  following  theorem,  which  follows  immediately  from 
§56: 


§61  CONTINUOUS  SERIES  49 

Theorem.  In  any  linear  continuous  series,  every  element  a  (unless 
it  be  the  first)  determines  a  fundamental  segment  (§  46)  of  the  so-called 
rational  elements,  namely,  the  series  of  all  the  rationals  ^preceding  a; 
and  conversely,  every  fundamental  segment  of  rationals  determines  an 
element  of  the  given  series,  namely,  the  upper  limit  of  the  segment  (§  56). 

The  rational  elements  of  the  given  series  correspond  to  the  fun- 
damental segments  which  have  limits  in  the  series  of  rationals;  the 
irrational  elements  correspond  to  the  segments  which  have  no 
limits  in  the  series  of  rationals  (§§  49,  50).  The  denumerable  dense 
series  considered  in  the  preceding  chapter  are  not  continuous,  since, 
as  we  have  seen  in  §  50,  they  contain  fundamental  segn^f^iits  which 
have  no  limits;  the  theorem  thus  brings  out  clearlj'-  tlit  sense  in 
which  the  linear  continuous  series  are  ''  richer  "  in  elements  than 
the  denumerable  dense  series. 

61.  The  type  6.  The  linear  continuous  series,  like  the  discrete 
series  or  the  denumerable  dense  series,  can  be  divided  into  four 
groiM)s,  distinguished  by  the  presence  or  absence  of  extreme  ele- 
ments; all  the  series  of  any  one  group  are  ordinally  similar  (see 
below),  and  therefore  constitute  a  definite  type  of  order.  In  parti- 
cular, a  linear  continuous  series  (§  54)  which  has  both  a  first  and  a  last 
element  is  called  by  Cantor  a  series  of  the  type  6,  or  the  type  of  the 
linear  continuum* 

The  proof  that  any  two  series  of  type  6  are  ordinally  similar 
follows  readily  from  the  analogous  theorem  in  regard  to  series  of 
type  r]  (§  45).*  For,  by  §  59  each  of  the  given  series  of  type  6  will 
contain  a  subclass  of  "  rational  "  elements  of  type  77;  by  §  45  these 
subclasses  of  rationals  can  be  brought  into  ordinal  correspondence 
with  each  other;  and  by  §  60  every  element  (except  the  first)  of 
each  of  the  given  series  is  uniquely  determined  as  the  limit  of  a 
fundamental  segment  of  rationals.  '^ 

It  should  be  noticed,  however,  that  this  correspondence  can  be 
set  up  in  an  infinite  number  of  ways,  since  not  only  the  selection  of 
rational  elements  from  the  given  series,  but  also  the  correspondence 
between  the  two  sets  of  rational  elements,  can  be  determined  in  an 
infinite  number  of  ways. 

*  G.  Cantor,  loc.  cit.  (1895),  §  11,  p.  511.    Russell,  loc.  cit.,  chap.  36. 


50  TYPES  OF  SERIAL  ORDER  §62 

62.  Since  the  definition  of  the  type  6  here  adopted  differs  in 
manner  of  approach,  though  not  in  substance,  from  the  definition 
given  by  Cantor,  I  add,  in  this  section,  a  statement  of  Cantor's 
definition  in  its  original  form.* 

Every  progression  or  regression  which  belongs  to  a  given  series  is 
called  by  Cantor  a  fundamental  sequence  (Fundamentalreihe) ;  any 
element  which  is  the  limit  of  any  fundamental  sequence  (upper 
limit  in  the  case  of  a  progression,  lower  limit  in  the  case  of  a  regres- 
sion), is  called  a  principal  element  (Hauptelement)  of  the  series. f  If 
every  fundamental  sequence  which  exists  in  a  given  series  has  a 
limit  in  the  series,  the  series  is  said  to  be  closed  (abgeschlossen) ;  if 
every  element  of  the  series  is  the  limit  of  some  fundamental  se- 
quence, the  series  is  said  to  be  dense-in-itself  (insichdicht) ;  and  any 
series  which  is  both  dense-in-itself  and  closed  is  said  to  be  perfect 
(perfekt) .  Finally,  if  a  series  is  such  that  between  any  two  elements 
there  are  other  elements,  the  series  is  said  to  be  dense  (uberalldicht) . 

The  following  theorems  follow  at  once  from  these  definitions: 

(1)  If  a  series  is  closed,  it  will  satisfy  Dedekind's  postulate 
(§54). 

(2)  If  a  series  satisfies  Dedekind's  postulate,  and  has  both  ex- 
treme elements,  it  will  be  closed. 

On  the  other  hand,  the  following  facts  should  be  noticed : 

(3)  A  series  may  satisfy  Dedekind's  postulate,  and  still  not  be 
closed,  as  witness  the  series  of  all  integers,  or  the  series  of  all  real 
numbers.! 

(4)  A  series  may  be  perfect  (that  is,  dense-in-itself  and  closed), 
and  not  be  dense ;  as  witness  the  series  discussed  in  §  52,  3  (with 
end-points),  or  the  series  of  all  real  numbers  from  0  to  3  inclusive 
with  the  omission  of  those  between  1  and  2. 

*  G.  Cantor,  loc.  cit.  (1895),  §§  10-11,  p.  508.  An  earlier  definition  of  the 
arithmetical  continuum  given  by  Cantor  in  Math.  Ann.,  vol.  5  (1872),  p.  123 
[cf.  ibid.,  vol.  21  (1883),  pp.  572-576],  involved  extra-ordinal  considerations, 
and  need  not  concern  us  here. 

t  This  definition  of  a  fundamental  sequence  is  inaccurately  quoted  by 
Veblen  {loc.  cit.,  p.  171),  who  leaves  out  the  regressions.    Thus,  in  the  series 

2',  1';  .  .  .  ,  -3,  -2,  -1,  0,  n,  +2,  +3,  .  .  .;  1",  2" 
the  element  1'  would  be  a  principal  element  according  to  Cantor's  definition, 
but  not  according  to  Veblen's.    [The  same  word,  Fundamentalreihe,  has  been 
used  by  Cantor  in  another  connection,  in  discussing  u-rg.tional  numbers;  Math. 
Ann.,  vol.  21  (1883),  p.  567]. 

%  It  is  therefore  perhaps  unfortunate  to  speak  of  Dedekind's  postulate  as  the 
postulate  of  closure.  ,    z^    / 


§62  CONTINUOUS  SERIES  51 

(5)  A  series  may  be  dent^e-in-itsolf  and  dense,  and  not  be  closed, 
as  for  example  the  series  of  rational  numbers  (with  or  without 
extreme  elements). 

(6)  A  series  may  be  dense  and  closed  and  not  be  dense-in-itself,* 
as  for  example  the  series  V  -\-  0  -{-  *V,  where  V  denotes,  for  the 
moment,  Veblen's  series  described  in  §  64,  3,  b,  and  *V  the  same 
series  in  reverse  order.  Here  the  element  0  is  not  the  limit  of  any 
fundamental  sequence,  since  every  progression  in  V  has  a  limit  in 
V,  and  every  regression  in  *F  has  a  limit  in  *V,  if  we  admit  the  valid- 
ity of  Cantor's  reasoning  in  regard  to  the  transfinite  well-ordered 
series  (§  83). 

(7)  A  series  may  be  perfect  (that  is,  dense-in-itself  and  closed), 
and  yet  have  no  last  element  and  no  first  element,  as  for  example 
the  series  *V  +  V.  Here  V  and  *V  have  the  meanings  just 
explained.! 

By  the  aid  of  these  definitions,  Cantor  defines  a  series  of  type  6 
by  the  following  two  conditions : 

(A)  the  series  must  be  perfect  (that  is,  dense-in-itself  and  closed); 
and 

(B)  the  series  must  contain  a  denumerable  subclass  R  in  such  a 
way  that  between  any  two  elements  of  the  given  series  there  is  an  ele- 
ment of  R. 

Every  series  which  satisfies  condition  B  will  clearly  be  dense. 

The  agreement  between  this  definition  and  that  given  in  §  61 
may  be  readily  established  by  the  reader.  The  use  of  Dede- 
kind's  postulate  instead  of  the  postulate  of  closure  implies  the  use 
of  fundamental  segments  instead  of  the  fundamental  sequences; 
this  modification  of  Cantor's  method  seems  to  me  desirable,  since 
every  segment  determines  a  unique  element,  and  every  element 
determines  a  unique  segment,  while  in  the  case  of  the  sequences, 
although  every  sequence  determines  a  unique  element,  it  is  not  true 
that  every  element  determines  a  unique  sequence.  J  I  have  pre- 
ferred Dedekind's  postulate  to  the  postulate  of  §  57  merely  because 
of  its  greater  symmetry. 

*  Compare  a  question  raiBed  by  Russell,  loc.  cit.,  p.  300.  The  series  given  in 
the  footnote  on  the  preceding  page  is  a  closed  series  which  is  neither  dense  nor 
dense-in-itself. 

t  Cf.  Hans  Hahn,  Monatsheftefur  Math,  und  Phys.,  vol.  21  (1910),  Literatur- 
herichte,  p.  26. 

J  It  can  be  shown,  however,  that  the  class  of  fimdamental  sequences  in  any 
continuous  series  has  the  same  "  cardinal  number  "  (§  88)  as  the  class  of  ele- 
ments in  the  series  itself  (compare  §  71).  ^'L 


52  '  TYPES  OF  SERIAL  ORDER  §  62a 

62a.  To  avoid  possible  confusion  with  §  62,  it  may  be  well  to  men- 
tion here  the  definitions  of  some  of  the  terms  used  in  the  theory  of 
sets  of  points,*  which  is  closely  related  to  the  theory  of  series. 

A  (linear)  set  of  points  is  a  collection  of  points  selected  in  any 
manner  from  the  points  of  a  straight  line.  Any  point  P  of  the 
line  is  called  a  cluster-point  (limit-point,  point  of  condensation)  of  the 
set,  if  in  every  interval  which  contains  P  as  an  interior  point  there 
are  points.of  the  set.  A  cluster  point  may  or  may  not  belong  to  the 
set.  A  set  is  called  closed  (abgeschlossen)  if  every  cluster  point  of 
the  set  belongs  to  the  set.  A  set  is  called  dense-in-itself  if  every 
point  of  the  set  is  a  cluster  point  of  the  set.  A  perfect  set  is  one 
which  is  both  closed  and  dense-in-itself.  A  set  is  called  every- 
ivhere-dense  if  between  every  two  distinct  points  of  the  line  there 
are  points  of  the  set. 

A  set  can  be  perfect  and  nowhere  dense,  as,  for  example,  the 
set  described  in  §  52,  3.  Every  perfect  set  can  be  put  into  one-to- 
one  correspondence  (sacrificing  order)  with  the  set  of  elements  in 
a  linear  continuum. 

The  derived  set  (Ahleitung)  of  a  given  set  is  the  set  composed  of 
all  the  cluster  points  of  the  given  set.  In  the  case  of  a  perfect 
set,  the  derived  set  is  the  same  as  the  given  set. 

A  set  is  called  compact  if  every  subclass  in  the  set  has  a  cluster 
point  in  the  set.  (Contrast  Russell's  use  of  "compact";  §  41, 
footnote.) 

Examples  of  linear  continuous  series 

63.  The  following  examples  serve  to  estabhsh  the  consistency 
of  the  postulates  of  the  present  chapter  (§  54;  compare  §  19) ;  in  all 
but  the  first  of  them  we  avoid  making  any  appeal  to  geometric 
intuition. 

(1)  The  simplest  geometric  example  of  a  linear  continuous  series 
is  the  series  of  all  points  on  a  line,  already  considered  in  §  55. 

The  most  important  non-geometrical  examples  are : 

(2)  The  class  of  (absolute)  real  numbers,  arranged  in  the  usual 
order;  and 

(3)  The  class  of  all  real  numbers  (positive,  negative,  and  zero), 
arranged  in  the  usual  order. 

*  For  references  to  recent  work  in  this  field  see  R.  E.  Root,  Trans.  Amer. 
Math.  Soc,  vol.  15  (1914),  pp.  51-71;  some  of  the  standard  treatises  are  men- 
tioned in  a  footnote  under  §  73. 


§63  CONTINUOUS  SERIES  53 

By  the  absolute  real  numbers  we  mean  the  class  of  all  fundamental 
segments  (§  46)  in  the  series  of  absolute  rational  numbers  (§51,2); 
and  by  the  usual  order  within  this  class  we  mean  that  a  segment  a 
shall  precede  a  segment  b  when  a  is  a  part  of  b* 

This  system  clearly  satisfies  the  general  conditions  for  a  series 
(§  12),  since  if  a  and  b  are  any  two  distinct  fundamental  segments 
of  any  dense  series,  one  of  them  must  be  a  part  of  the  other,  and 
the  relation  of  inclusion  is  transitive.  Further,  the  series  is  dense; 
for,  if  a  segment  a  is  part  of  a  segment  b,  there  will  always  be 
rationals  belonging  to  b  and  not  to  a;  a  segment  x  containing  the 
segment  a  and  some  of  these  rationals  will  then  lie  "  between  "  the 
segments  a  and  b.  To  show  that  Dedekind's  postulate  is  also 
satisfied,  suppose  that  the  whole  series  K  is  divided  in  any  way  into 

*  This  is  the  definition  adopted  by  RusseU  {loc.  cit.,  chap.  33);  it  was  first 
given  in  this  form  bj^  M.  Pasch  {Differential-  und  Integralrechnung,  1882),  his 
Zahlenstrecke  (fundamental  segment  of  rationals)  being  a  modification  of 
Dedekind's  Schnitt  or  cut  (1872).  Similar  definitions  have  been  given  by 
Dedekind  (1872),  Cantor  (1872),  Peano  (1899),  and  others;  a  historical  ac- 
count is  given  by  Peano  in  Rev.  de  Math.,  vol.  6  (1899),  pp.  126-140.  The  con- 
struction of  the  system  of  (absolute)  real  numbers  may  be  briefly  described  as 
follows  (confining  ourselves  to  the  positive  numbers):  (1)  the  integers  are  the 
natural  numbers,  assumed  as  known;  (2)  the  rationals  are  pairs  of  integers; 
and  (3)  the  reals  are  classes  (fundamental  segments)  of  rationals.  As  a  matter 
of  convenience  in  notation,  a  pair  of  integers  in  which  the  denominator  is  1  is 
represented  by  the  numerator  alone;  rational  numbers  of  this  form  are  said  to 
be  integral,  while  aU  other  rational  numbers  are  caUed  fractional.  Again,  a  fun- 
damental segment  which  has  a  limit  in  the  series  of  rationals  is  represented  by 
the  same  symbol  as  its  limit;  real  numbers  of  this  form  are  said  to  be  rational, 
while  all  other  real  numbers  are  called  irrational  (compare  §  50).  This  notation, 
however,  should  not  be  interpreted  as  meaning  that  the  class  of  real  nmnbers 
includes  the  class  of  rationals,  or  that  the  class  of  rational  numbers  includes  the 
class  of  integers.  On  the  contrary,  while  the  "  integral  number  2  "  means 
simply  the  second  number  in  the  natural  series,  the  "  rational  number  2  " 
means  the  pair  of  natural  numbers  2  and  1,  and  "  the  real  number  2  "  means 
the  class  of  all  rational  numbers  which  precede  the  rational  number  2/1.  The 
rules  by  which  the  sum  and  product  of  two  real  nimabers  are  defined  do  not 
concern  us,  in  this  discussion  of  the  purely  ordinal  theory;  see  O.  Stolz  and 
J.  A.  Gmeiner,  Theoretische  Arithmetik  (1901-  );  J.  Tannery,  Introduction  a  la 
thearie  desfonctions  (2nd  edit.,  1904);  H.  Weber  and  J.  Wellstein,  Encyclopddie 
der  Elementar-Mathematik  (vol.  1,  1903);  E.  V.  Huntington,  Trans.  Amer. 
Math.  Soc,  vol.  6  (1905),  pp.  209-229,  or  the  two  monographs  cited  in  the 
introduction;  A.  Loewy,  Lehrbvjch  der  Algebra  (1915). 


54  TYPES  OF  SERIAL  ORDER  §63 

two  parts  Ki  and  K^  such  that  every  element  of  Ki  precedes  every 
element  of  K2;  then  the  class  of  all  rationals  which  belong  to 
any  element  of  Ki  will  be  a  fundamental  segment  in  the  series  of 
rationals,  and  will  be  the  element  X  demanded  in  the  postulate. 
Fmally,  the  series  is  a  linear  continuous  series,  since  we  may  take 
as  the  required  subclass  R  all  the  elements  of  K  which  have  limits 
in  the  series  of  rationals  (§  49). 

By  the  series  of  all  real  numbers  (positive,  negative,  or  zero)  we 
then  mean  a  series  built  up  from  the  series  of  absolute  real  numbers 
in  the  same  way  as  the  series  of  all  rationals  was  built  up  from 
the  series  of  absolute  rationals  in  §  51,  3.  Or  again,  all  real  num- 
bers may  be  defined  as  fundamental  segments  of  the  series  of  all 
rationals,  just  as  the  absolute  real  numbers  are  defined  as  funda- 
mental segments  of  the  series  of  absolute  rationals. 

In  the  series  of  real  numbers  we  have  thus  constructed  an  arti- 
ficial system  which  certainly  satisfies  all  the  conditions  for  a  linear 
continuous  series  (§  54) ;  there  can  therefore  be  no  doubt  that  those 
conditions  are  free  from  inconsistency.*  If  we  assume  as  geometri- 
cally evident  that  the  series  of  all  points  on  a  line  an  inch  long  also 
satisfies  these  conditions,  then  an  ordinal  correspondence  can  be 
established  between  the  real  numbers  and  the  points  of  the  line,  in 
accordance  with  §  61  (taking  as  the  "  rational  "  points  of  the  line 
those  points  whose  distances  from  one  end  of  the  line  are  proper 
fractions  of  an  inch);  but  in  setting  up  this  correspondence  we 
must  recognize  that  the  continuity  of  the  series  of  points  on  the  line 
is  an  assumption  which  is  not  capable  of  direct  experimental  veri- 
fication. 

(4)  Another  example  of  a  linear  continuous  series  is  the  class  of 
aU  non-terminating  decimal  fractions,  arranged  in  the  usual  order 
(§  19,  9;  §  40). 

This  series  is  dense;  for,  suppose  a  and  b  are  any  two  of  the 
decimals  such  that  a  <  b;  let  /S^  be  the  first  digit  of  b  which  is 
greater  than  the  corresponding  digit  of  a,  and  let  ^n  be  the  first 

*  Cf.  H.  Weber,  Algebra,  vol.  1,  p.  7,  where  the  real  numbers  are  defined 
(after  Dedekind)  as  "  cuts  "  in  the  series  of  rationals,  instead  of  as  fundamental 
segments  of  rationals.  (A  cut  is  simply  a  rule  for  dividing  a  series  K  into  two 
non-empty  parts  Ki  and  K2,  such  that  every  element  of  Ki  precedes  every  ele- 
ment of  K2,  while  Ki  and  Kt  together  exhaust  the  series  K.) 


§G4  CONTINUOUS  SERIES  55 

digit  beyond  /S^  which  is  different  from  0;  then  any  decimal  x  in 
which  the  first  n  —  1  digits  are  the  same  as  in  6,  while  the  nth  digit 
is  less  by  one  than  i3„,  will  lie  between  a  and  h.  Furth*.  r.  the  series 
satisfies  Dedekind's  postulate;  for,  if  Ki  and  K2  are  the  given  sub- 
classes, we  may  determine  the  decimal  X  =  .^1^21.3  ...  as  fol- 
lows: ^1  is  the  largest  digit  which  occurs  in  the  first  place  of  any 
decimal  belonging  to  Ki ;  ^2  is  the  largest  digit  which  occurs  in  the 
second  place  of  any  decimal  beginning  with  ^1  and  belonging  to  Ki ; 
^3  is  the  largest  digit  which  occurs  in  the  third  place  of  any  decimal 
beginning  with  ^1^2  and  belonging  to  Ki;  and  so  on.  Finally,  the 
series  is  linear,  since  we  may  take  as  the  subclass  R  the  class  of 
thpse  decimals  in  which  all  the  places  after  any  given  place  are 
filled  with  9's.  —  The  series,  as  we  notice,  contains  a  last  element 
(.999  .  .  .),  but  no  first. 

(5)  As  a  final  example  we  mention  the  series  described  in  §  19,  8, 
namely :  K  =  the  class  of  all  possible  infinite  classes  of  the  natural 
numbers,  no  number  being  repeated  in  any  one  class;  with  the 
relation  <  so  defined  that  a  <  b  when  the  smallest  number  in  a  is 
less  than  the  smallest  number  in  b,  or,  if  the  smallest  n  numbers  of 
a  and  6  are  the  same,  when  the  (n  -\-  l)st  number  of  a  is  less  than 
the  (n  -\-  l)st  number  of  b. 

This  series  is  continuous,  as  the  reader  may  readily  verify;  and 
it  may  be  shown  that  it  satisfies  the  postulate  of  linearity,  since  we 
may  take  as  the  subclass  R  the  class  of  all  the  elements  in  which 
only  a  finite  number  of  the  natural  numbers  are  absent.  We  notice 
also  that  the  series  contains  a  first  element  (namely  the  class  of  all 
the  natural  numbers),  but  no  last  element. 

This  example  is  particularly  interesting  as  showing  how  a  linear 
continuous  series  can  be  built  up  directly  from  the  natural  numbers, 
without  making  use  of  the  rationals.* 

Examples  of  series  which  are  not  linear  continuous  series 

64.  The  examples  given  in  this  section  serve  to  show  (compare 
§  20)  that  postulates  CI  and  C2  (§  54)  are  independent  of  each 
other,  and  that  postulate  C3  is  independent  of  both  of  them. 
Postulate  C2,  on  the  other  hand,  is  clearly  a  consequence  of  postu- 
late C3. 

*  B.  Russell,  Principles  of  Mathematics,  vol.  1,  p.  299. 


56  TYPES  OF  SERIAL  ORDER  §64 

(1)  Dense  series  which  do  not  satisfy  Dedekind's  postulate. 

(a)  Denumerable  series  which  are  dense  but  do  not  satisfy 
Dedekind's  postulate  are  given  in  §  51. 

(6)  A  non-denumerable  example  of  the  same  sort  is  the  series  of 
all  the  points  on  a  line  with  the  exception  of  some  single  point;  or 
better,  the  series  described  in  §  52,  2,  b. 

(2)  Series  which  satisfy  Dedekind^s  postulate,  but  are  not  dense. 

(a)  The  series  described  in  §  52,  3  (consisting  of  the  ternary- 
fractions  in  which  the  digits  0  and  2  only  are  used)  is  not  dense,  but 
can  readily  be  shown  to  satisfy  the  postulate  of  Dedekind. 

(6)  Any  discrete  series  is  also  an  example  of  this  kind. 

(3)  Continuous  series  which  are  not  linear. 

(a)  Let  K  be  the  class  of  all  couples  (x,  y),  where  x  and  y  are  real 
numbers  from  0  to  1  inclusive;  and  let  {xi,  yi)  <  {x2,  y^)  when 
Xi  <  X2,  or  when  Xi  =  x^  and  yi  <  y^.  This  series  is  a  continuous 
series  (satisfying  CI  and  C2) ;  but  it  is  not  a  linear  continuous  series, 
since  no  denumerable  subclass  R  of  the  kind  demanded  in  postulate 
C3  is  possible  within  it.  (The  same  example,  in  geometric  form, 
has  been  mentioned  already  in  §  55;  other  examples  of  a  similar 
kind  will  occur  in  §  70.) 

(6)  Let  coi  (or  Q)  be  the  smallest  of  the  well-ordered  series  of 
Cantor's  third  class  (see  §83,  below),  and  connect  each  element 
with  the  next  following  element  by  a  linear  continuous  series;  the 
resulting  series,  which  has  been  proposed  by  Veblen,*  is  continuous 
but  contains  no  denumerable  subclass  R  of  the  kind  demanded  in 
postulate  C3,  since  every  denumerable  subclass  in  the  series  has 
an  upper  limit  in  the  series  (cf.  §  85). 

(4)  A  series  which  is  not  continuous  and  not  dense. 

As  a  final  example  of  a  series  which  is  not  continuous,  we  men- 
tion a  class  K  composed  of  two  sets  of  real  numbers,  say  red  and 
blue,  with  a  relation  of  order  defined  as  follows:  of  two  elements 

*  O.  Veblen,  Trans.  Amer.  Math.  Soc,  vol.  6  (1905),  p.  169.  Another  in- 
teresting series  may  be  made  from  the  series  i2  by  connecting  each  element 
with  the  next  foUowing  element  by  a  series  of  type  ■n',  this  series  is  dense  and 
dense-in-itself  but  not  denumerable  and  not  closed  (cf.  §  62,  5). 


§65  CONTINUOUS  SERIES  57 

which  have  unequal  numerical  values,  that  one  shall  precede 
which  would  precede  in  the  usual  order  of  real  numbers,  regardless 
of  color;  of  two  elements  which  have  the  same  numerical  value,  the 
red  shall  precede. 

This  system  is  built  up  by  interpolating  the  elements  of  one  con- 
tinuous series  between  the  elements  of  another  continuous  series; 
the  resulting  series,  instead  of  being  "more  continuous"  as  one 
might  have  been  tempted  to  expect,  is  no  longer  even  dense,  since 
every  red  element  has  an  immediate  successor  (compare  §  52,  l,b). 

Arithmetical  operations  among  the  elements  of  a  continuous  series 

65.  In  the  case  of  continuous  series  as  in  the  case  of  dense  series 
it  is  not  possible  to  give  purely  ordinal  definitions  of  the  sums  and 
products  of  the  elements;  for,  unless  some  other  fundamental 
notion  besides  the  notion  of  order  is  introduced,  the  elements  of 
these  series  (except  extreme  elements)  have  no  ordinal  properties 
by  which  we  can  tell  them  apart  (compare  §  53) .  We  might,  to  be 
sure,  define  sums  and  products  of  the  elements  of  some  particular 
series  (like  the  series  of  real  numbers,  in  the  usual  order)  by  the  use 
of  extra-ordinal  properties  peculiar  to  that  series,  and  then  transfer 
these  definitions  to  other  series  of"  the  same  type  by  a  one-to-one 
ordinal  correspondence;  but  this  method  would  be  wholly  inade- 
quate, since  the  ordinal  correspondence  could  be  set  up  in  an 
infinite  number  of  ways.  To  construct  a  completely  determinate 
continuous  system  it  is  therefore  necessary  to  introduce  some 
further  notions,  like  addition  and  multiplication,  besides  the  notion 
of  order,  as  fundamental  notions  of  the  system.* 

*  See  for  example  my  set  of  postulates  for  ordinary  complex  algebra,  Trans. 
Amer.  Math.  Soc,  vol.  6  (1905),  pp.  209-229,  especially  §  8,  or  my  monograph 
on  The  Fundamental  Propositions  of  Algebra,  cited  in  the  introduction;  or  my 
postulates  for  absolute  continuous  magnitude,  Trans.  Amer.  Math.  Soc, 
vol.  3  (1902),  pp.  264-279. 


CHAPTER  VI 

Continuous  Series  of  More  than  One  Dimension, 
WITH  A  Note  on  Multiply  Ordered  Classes 

66.  In  the  preceding  chapters  we  have  studied  various  kinds  of 
series,  or  simply  ordered  classes  (§12), — especially  the  linear 
continuous  series  (§  54).  In  the  foMowing  chapter  we  consider 
briefly  some  kinds  of  continuous  series  which  are  not  linear,  and 
add  a  short  note  on  multiply  ordered  classes. 

Continuous  series  of  more  than  one  dimension* 

67.  We  shall  use  the  term  one-dimensional  framework  or  skeleton 
(Ri)  to  denote  a  series  of  type  rj,  that  is,  a  denumerable  dense  series 
without  extreme  elements  (§  44) .  A  one-dimensional,  or  linear, 
continuous  series  is  then  any  continuous  series  which  contains  a 
framework  Ri  in  such  a  way  that  between  any  two  elements  of  the 
given  series  there  are  elements  of  ^i  (§  59). 

Again,  a  two-dimensional  frarnework,  R^,  is  any  series  formed 
from  a  one-dimensional  continuous  series  by  replacing  each  element 
of  that  series  by  a  series  of  type  -q ;  and  a  two-dimensional  co7itinu- 
ous  series  is  any  continuous  series  which  contains  a  framework  R2 
in  the  same  way. 

And  so  on.  In  general,  an  n-dimensional  framework,  Rn,  is  any 
series  formed  from  an  (n  —  1) -dimensional  continuous  series  by 
replacing  each  element  of  that  series  by  a  series  of  type  17;  and  an 
n-dimensional  continuous  series  is  any  continuous  series  which 
contains  a  framework  Rn  in  such  a  way  that  between  any  two  ele- 
ments of  the  given  series  there  are  elements  of  Rn. 

*  The  study  of  the  multi-dimensional  continuous  series  was  proposed  by 
Cantor  in  Math.  Ann.,  vol.  21,  p.  590,  note  12  (1883),  but  seems  never  to  have 
been  carried  out  in  detail.  It  would  be  interesting  to  extend  the  discussion  to 
continuous  series  of  a  transfinite  number  of  dimensions  (cf.  §  88). 


§69  MULTIPLY  ORDERED  CLASSES  59 

68.  By  a  k-dimensional  section  of  any  continuous  series  we  shall 
mean  any  segment  (§  47)  which  forms  by  itself  a  /o-dimensional 
continuous  series,  but  is  not  a  part  of  any  other  such  segment.* 

In  an  n-dimensional   continuous  series  each   one-dimensional 

section,  unless  it  be  the  j^^^ ,  will  have  a  ^^^^^  element,  and  these 
elements  taken  in  order  will  form  an  (n  —  1) -dimensional  continu- 
ous series.    And  so  in  general :  each  /c-dimensional  section,  unless  it 

be  the  j^^^,  will  have  a  ^^^^  (k  —  1) -dimensional  section,  and  these 

(k  —  1) -dimensional  sections  taken  in  order  will  be  the  elements  of 
an  (n  —  /c) -dimensional  continuous  series. 

69.  As  already  noted,  there  are  four  different  types  of  one- 
dimensional  continuous  series,  distinguished  by  the  presence  or 
absence  of  extreme  elements;  in  particular,  a  one-dimensional 
continuous  series  with  both  a  first  and  a  last  element  is  called  a 
series  of  type  6  (§  61). 

A  two-dimensional  continuous  series  may  or  may  not  have  a  first 
one-dimensional  section,  and  that  section  in  turn  may  or  may  not 
have  a  first  element.  Similarly,  there  may  or  may  not  be  a  last 
one-dimensional  section,  which  in  turn  may  or  may  not  have  a  last 
element.  There  are  therefore  nine  different  types  of  such  series, 
distinguished  by  their  initial  and  terminal  properties.  In  particu- 
lar, a  two-dimensional  continuous  series  with  both  a  first  and  a  last 
element  we  may  caU  a  series  of  type  6^  (since  it  may  be  formed  from 
a  series  of  type  6  by  replacing  each  element  by  another  series  of 
type0).t 

And  so  on.  In  general,  there  will  be  (n  -f-  1)^  different  types  of 
n-dimensional  continuous  series,  distinguished  by  their  initial  and 
terminal  properties.  In  particular,  an  n-dimensional  continuous 
series  which  has  both  a  first  and  a  last  element  may  be  called  a 
series  of  type  6". 

*  We  may  speak  of  a  section  of  a  framework  R„,  as  well  as  of  a  section  of  a 
continuous  series.  A  "  zero-dimensional  "  section  would  be,  of  course,  a  single 
element.  —  If  preferred,  the  word  constituent  may  be  used  instead  of  section. 

t  Cf.  Cantor's  notation  for  the  "  product "  of  two  well-ordered  series 
(§86). 


3    t^Mi-e.     i' 


60  TYPES  OF  SERIAL  ORDER  §70 

The  proof  that  any  two  series  of  the  same  type  are  ordinally 

sunilar,  and  that  all  the  types  are  distinct,  is  readily  obtained  by  an 

extension  of  the  methods  used  in  §§45  and  61. 

-A-  70.   An  example  of  an  n-dimensional  continuous  series  is  a  class 

'  "^  ,  whose  elements  are  sets  of  real  numbers  (xi,  Xi,  Xz,  .  .  .  ,  x„),  where 

f .  ^  -  ^  '^^'^ '  /  i  \^^  '^^  ^^y  ^^^^  number,  and  Xi,  Xz,  .  .  .  ,  Xn  are  restricted  to  the 

i-ii  )H  /   interval  from  0  to  1  inclusive ;    the  elements  of  the  class  being 

-fu    ':l^"^^t^^  arranged  primarily  in  order  of  the  a^i's;  or  in  case  of  equal  XiS,,  in 

ir^ '  P^w'^     order  of  the  rr2's;  or  in  case  of  equal  a^i's  and  equal  X2's,  in  order  of 

■^  Lr^-'^     the  rcs's ;  etc. 

^  f.    f^Ua^   ,        If  n  =  1,  2,  or  3,  the  elements  of  this  class  can  be  represented 

i  cUav.  '  ^^" '  geometrically:   (1)  by  the  points  on  a  line;   (2)  by  the  points  of  a 

^   ,  ^AAs-^^~    '^^  plane  region  bounded  by  two  parallel  lines;  and  (3)  by  the  points 

Uf  1^  ^^"  of  a  space  region  bounded  by  a  square  prismatic  surface.    If  n  is 

greater  than  3,  no  simple  geometrical  interpretation  is  possible. 

71.  Although  the  various  types  of  series  just  considered  are  all 
distinct  as  types  of  order,  yet  it  is  important  to  notice  that  the  class 
of  elements  of  an  n-dimensional  continuous  series  can  be  put  into 
one-to-one  correspondence  with  the  class  of  elements  of  a  one-dimen- 
sional continuous  series,  if  the  relation  of  order  is  sacrificed ;  or,  in 
the  terminology  of  modern  geometry,  the  points  of  all  space  (of  any 
number  of  dimensions)  can  be  put  into  one-to-one  correspondence  with 
the  points  of  a  line.  One  of  Cantor's  most  interesting  early  dis- 
coveries was  a  device  for  actually  setting  up  this  correspondence; 
we  give  a  sketch  of  the  method  for  the  case  of  two  dimensions.* 

As  a  preliminary  step,  we  notice  that  a  one-to-one  correspond- 
ence can  be  set  up  between  the  points  of  any  two  lines,  of  length  a 
and  6,  with  or  witho'ut  end-points.  For,  each  line  can  be  divided 
into  a  denumerable  set  of  segments  of  lengths  equal,  say,  to  ^,  |, 
I,  ...  of  the  length  of  the  line;  a  one-to-one  correspondence  can 
be  established  between  the  two  sets  of  segments,  and  then  (as  in 
§  3)  between  the  interior  points  of  each  segment  of  one  set  and  the 
interior  points  of  the  corresponding  segment  of  the  other  set;  and  a 
one-to-one  correspondence  can  also  be  established  between  the  two 
sets  of  points  of  division. 

*  Cantor,  Crelle's  Journ.  fur  Math.,  vol.  84,  pp.  242-258  (1877);  cf.  Math. 
Ann.,  vol.  46,  p.  488  (1895). 


§71  MULTIPLY  ORDERED  CLASSES  61 

Consider  now  the  points  {x,  y)  within  a  square  one  inch  on  a  side 
(0<a;<l,  0<y<l),  and  the  points  t  on  a  hnc  say  three  inches 
long  (0  <  ^  <  3) ;  and  divide  each  third  of  the  hne  t  into  a  denum- 
erable  set  of  segments  of  lengths  ^,  j,  I,  .  .  .  of  an  inch.  A  one-to- 
one  correspondence  between  the  points  of  the  square  and  the  points 
of  the  line  can  then  be  established  as  follows : 

(1)  The  points  {x,  y)  for  which  x  and  y  are  both  rational  form  a 
denumerable  set,  and  can  therefore  be  put  into  one-to-one  corre- 
spondence with  the  "  rational  "  points  of  the  line  —  that  is,  the 
points  for  which  t  is  rational. 

(2)  The  points  {x,  y)  for  which  x  is  rational  and  y  irrational  are 
the  "  irrational  "  points  of  a  denumerable  set  of  vertical  lines,  and 
can  therefore  be  put  into  one-to-one  correspondence  with  the 
"  irrational "  points  of  the  denumerable  set  of  segments  which 
occupies,  say,  the  last  third  of  the  line. 

(3)  Similarly  the  pomts  (x,  y)  for  which  y  is  rational  and  x  irra- 
tional can  be  put  into  one-to-one  correspondence  with  the  "  irra- 
tional "  points  of  the  middle  third  of  the  line. 

(4)  Finally,  the  points  for  which  x  and  y  are  both  irrational  can 
be  put  into  one-to-one  correspondence  with  the  "  irrational " 
points  of  the  first  third  of  the  line.  For,  every  irrational  number  a 
between  0  and  1  can  be  expressed  as  a  non-terminating  simple  con- 
tinued fraction,  a  =  [ai,  a^,  as,  .  .  .],  that  is: 

1 

ai-\ 


1 


^2  + 

as  +  .  .  . , 
where  ai,  02,  as,  .  .  .  are  positive  integers;  so  that  to  the  point 

X  =   [Xi,  X2,  Xs,    .    .    .J, 

y  =  [yi,  y-2,  yz,  ■  •  •] 

in  the  square  we  can  assign  the  point 

t  =  [xi,  yi,  xi,  yi,  Xz,  t/3,  .  .  .  ] 
on  the  hne;  while  inversely,  to  the  point 

i  =  1^1}  fej  ^3,  .  .  .  J 
on  the  hne  we  can  assign  the  point 

y  =  \h,  ti,  ^6,  .  .  •  ] 

in  the  square. 


'.,  jL 


62  TYPES  OF  SERIAL  ORDER  §72 

Thus  the  correspondence  between  the  points  of  the  square  and 
the  points  of  the  hne  is  complete;  and  the  method  is  easily  extended 
to  any  number  of  dimensions,  finite  or  denumerably  infinite. 

Note  on  multiply  ordered  classes 

72.  A  multiply  ordered  class  is  a  system  (§  1 1)  consisting  of  a  class 
K  the  elements  of  which  may  be  ordered  according  to  several  differ- 
ent serial  relations. 

For  example,  a  class  of  musical  tones  may  be  arranged  in  order 

according  to  pitch,  or  according  to  intensity,   or  according  to 

duration.    Again,  the  class  of  points  in  space  may  be  ordered  in 

various  ways  according  to  their  distances  from  three  fixed  planes. 

A  multiply  ordered  class  may  also  be  called  a  multiple  series;  but 

■-^^^  a  system  of  this  kind  is  not  strictly  a  series  with  respect  to  any  one 

"^  of  its  ordering  relations,  since  postulate  1  does  not  strictly  hold 

* (see  §  12  or  §  74).    A  multiple  series  which  is  of  type  0  with  respect 

"Pr^  ^^, .      I  to  each  of  n  serial  relations  is  called  an  n-dimensional  continuum. 
An  extended  discussion  of  multiply  ordered  classes  is  contained 
in  Cantor's  memoir  of  1888.* 

*  Cantor,  Zeitschr.  f.  Phil.  u.  philos.  Kritik,  vol.  92,  pp.  240-265  (1888). 
See  also  F.  Riesz,  Math.  Ann.,  vol.  61,  pp.  406-421  (1905). 


J ' 


CHAPTER  VII 

Well-Ordered  Series,  with  an  Introduction  to 
Cantor's  Transfinite  Numbers 

73.  In  §§  21,  41,  and  54,  certain  special  kinds  of  series  {"  dis- 
crete," "  dense,"  "  continuous  ")  have  been  defined,  and  their 
chief  properties  discussed. 

In  this  chapter  a  brief  account  is  now  to  be  given  of  another 
special  kind  of  series,  which  has  proved  to  be  of  fundamental  im- 
portance in  Cantor's  theory  of  the  transfinite  numbers,  and  I  hope 
that  some  readers  may  be  led,  by  this  brief  introduction,  to  a 
further  study  of  that  most  recent  development  of  mathematical 
thought,  in  which  many  problems  of  fundamental  interest  still 
await  solution. 

The  theory  of  the  transfinite  numbers  was  created  by  Georg 
Cantor  in  1883,  in  a  monograph  called  Grundlagen  einer  allgemeinen 
Mannichfaltigkeitslehre;  ein  mathematisch-philosophischer  Versuch 
in  der  Lehre  des  Unendlichen.  A  much  clearer  presentation  of  the 
subject  will  be  found  in  his  Beitrdge  zur  Begrilndung  der  transfiniten 
Mengenlehre  in  the  Mathematische  Annalen  (1895,  1897)  translated 
by  P.  E.  B.  Jourdain,  Contributions  to  the  Founding  of  the  Theory  of 
Transfinite  Numbers  (Open  Court  Pub.  Co.,  1915) ;  but  many  of  the 
speculations  which  were  begun  or  suggested  in  the  Grundlagen  have 
not  yet  been  developed.* 

*  Among  the  more  recent  treatises  may  be  mentioned:  A.  Schonflies, 
Entwickelung  der  Mengenlehre  und  ihrer  Anwendungen,  second  edition,  1913 
(Teubner,  Leipzig);  B.  Russell,  Principles  of  Mathematics  (1903);  L.  Cou- 
turat,  Les  Principes  des  mathematiques  (1905);  G.  Hessenberg,  Grundbegriffe 
der  Mengenlehre  (1906);  W.  H.  and  G.  C.  Yoimg,  The  Theory  of  Sets  of  Points 
(1906);  J.  Konig,  Netie  Grundlagen  der  Logik,  Arithmetik  und  Mengenlehre 
(1914);  F.  Hausdorff,  Grundziige  der  Mengenlehre  (1914);  P.  E.  B.  Jourdain, 
The  Development  of  the  Theory  of  Transfinite  Numbers,  published  serially  in 
Archiv  der  Math.  u.  Phys.,  ser.  3,  volumes  10,  14,  16,  22  (1906-1913);  and  the 
Principia  Mathematica  by  Whitehead  and  Russell,  vol.  3  (1913). 

63 


64  TYPES  OF  SERIAL  ORDER  §74 

74.  A  series,  or  simply  ordered  class,  has  been  defined  in  §  12  as 
any  system  {K,  < )  which  satisfies  the  following  three  conditions : 

Postulate  1.  If  a  and  b  are  distinct  elements  of  the  class  K,  then 
either  a  <  h  or  h  <  a. 

Postulate  2.   If  a  <  b,  then  a  and  b  are  distinct. 

Postulate  3.   If  a  <  b  and  b  <  c,  then  a  <  c. 

A  normal  series,  or  "well-ordered"  series  (wohlgeordnete  Menge),* 
is  then  any  series  which  satisfies  the  following  three  conditions:  f 

*  The  earliest  of  Cantor's  writings  which  bear  upon  this  subject  will  be 
found  in  Malh.  Ann.,  vol.  5,  pp.  123-132  (1872);  and  in  Crelle's  (or  Bor- 
chardt's)  Journ.  fiir  Math.,  vol.  77,  pp.  258-262  (1874);  vol.  84,  pp.  242-258 
(1877).  Then  came  a  series  of  six  articles  "  tJber  unendhche,  lineare  Punkt- 
mannichfaltigkeiten,"  Math.  Ann.,  vol.  15,  pp.  1-7  (1879);  vol.  17,  pp.  355- 
358  (1880);  vol.  20,  pp.  113-121  (1882);  vol.  21,  pp.  51-58  (1883);  vol.  21, 
pp.  545-591  (1883);  vol.  23,  pp.  453-488  (1884).  The  fifth  of  these  articles  is 
identical  with  the  monograph  pubhshed  in  the  same  year  (1883)  under  the 
title  "  Grundlagen  einer  allgemeinen  Mannichfaltigkeitslehre  "  —  page  n  of 
the  "  Grundlagen  "  corresponding  to  page  (n  +  544)  of  the  article  in  the 
Annalen.  [AH  the  articles  mentioned  thus  far,  or  partial  extracts  from  them, 
are  translated  into  French  in  the  Acta  Mathematica,  vol.  2,  1883.  The  same 
journal  contains  also  some  further  contributions;  see  vol.  2,  pp.  409-414 
(1883);  vol.  4,  pp.  381-392  (1884);  vol.  7,  pp.  105-124  (1885).]  These  articles 
were  followed  by  a  number  of  writings  in  defence  of  the  new  theory;  see  espe- 
cially the  Zeitschnft  fur  Phil,  und  philos.  Kritik,  vol.  88,  pp.  224r-233  (1886); 
vol.  91,  pp.  81-125,  252-270  (1887) ;  vol.  92,  pp.  240-265  (1888).  Then  came  a 
short  but  interesting  note  in  the  Jahresber.  d.  D.  Math.-Ver.,  vol.  1,  pp.  75-78 
(1892),  and  finally  the  "  Beitrage,"  etc.,  Math.  Ann.,  vol.  46,  pp.  481-512 
(1895);  vol.  49,  pp.  207-246  (1897);  French  translation  by  F.  Marotte  (1899); 
English  translation  by  P.  E.  B.  Jourdain  (1915).  Since  1897  the  literature  of 
the  subject  has  rapidly  increased,  but  nothing  further  has  been  published  by 
Cantor  himself. 

t  G.  Cantor,  Math.  Ann.,  vol.  21  (1883),  p.  548;  ibid.,  vol.  49  (1897),  p.  207. 
The  name  "  normal  series "  was  suggested  to  me  by  the  term  "  normally 
ordered  class,"  used  by  E.  W.  Hobson  as  a  translation  of  wohlgeordnete  Menge; 
Proc.  Lond.  Math.  Soc,  ser.  2,  vol.  3  (1905),  p.  170.  It  would  have  been  a 
better  term  than  "well-ordered  series,"  for  the  adjective  "well-ordered" 
apphes  properly  only  to  a  class,  not  to  a  series,  since  a  series  is  already  an 
ordered  class,  and  a  well-ordered  class  would  be,  as  it  were,  a  "  well "  series. 
But  the  term  "  well-ordered  "  is  so  well  established  in  the  literature  that  it 
seems  best  to  retain  it  as  the  designation  for  this  particular  kind  of  series. 


§76  WELL-ORDERED  SERIES  65 

Postulate  4.    The  series  has  a  first  element  (§  17). 

Postulate  5.  Every  element,  unless  it  he  the  last,  has  an  imme- 
diate successor  (§  17). 

Postulate  6.  Every  fundamental  segment  of  the  series  has  a 
limit. 

Here  a  "  fundamental  segment  "  is  any  lower  segment  which 
has  no  last  element;  the  "limit"  of  a  fundamental  segment  is 
the  element  next  following  all  the  elements  of  the  segment 
(§§46,49). 

The  consistency  and  independence  of  these  postulates  are  estab- 
lished by  the  examples  already  given  in  §§  28-29. 

In  a  well-ordered  series,  any  element  which  is  the  limit  of  a  fun- 
damental segment  (and  therefore  has  no  immediate  predecessor)  is 
called  a  limiting  element  of  the  series  (Grenzelement,  Element  der 
zweiten  Art*).  Eveiy  element  which  is  neither  a  limiting  element, 
nor  the  first  element  of  the  series,  will  have  a  predecessor. 

For  example,  the  series 

li,  2i,  3i,  .  .  .;     I2,  22,  82,  .  .  .',    I3,  23,  03,  .  .  .J     .  .  .;  1 

is  a  well-ordered  series  in  which  the  limiting  elements  (I2,  I3,  .  .  .; 
1')  form  a  progression  followed  by  a  last  element  I'. 

75.  From  postulates  1-6  it  follows  at  once  that  Dedekind's 
postulate  (see  §  21  or  §  54)  will  hold  true  in  any  well-ordered  series; 
indeed  we  may  use  Dedekind's  postulate  in  place  of  postulate  6  in  the 
definition  of  a  well-ordered  series;'\  I  prefer  postulate  6  in  this  case, 
however,  because  it  emphasizes  the  unsymmetrical  character  of  the 
well-ordered  series. 

76.  Other,  very  convenient,  forms  of  the  definition  are  the 
following : 

(1)  A  well-ordered  series  is  any  series  in  which  every  subclass  (§6) 
has  a  first  element. t 

*  G.  Cantor,  Math.  Ann.,  vol.  49  (1897),  p.  226.    Jourdain  uses  Ldmes;  Phil. 
Mag.,  ser.  6,  vol.  7  (1904),  p.  296.     Compare  §  62,  above, 
t  O.  Veblen,  Trans.  Amer.  Math.  Soc.,  vol.  6  (1905),  p.  170. 
i  Cantor,  loc.  cit.  (1897),  p.  208. 


66  TYPES  OF  SERIAL  ORDER  §77 

■  '/u^t.  ^^^      (2)  A  well-ordered  series  is  any  series  which  contains  no  subclass  of 
'^^■^f^*   the  type  *aj;  that  is,  no  subclass  which  is  a  regression  (§  25).* 

The  equivalence  of  each  of  these  definitions  with  the  definition  in 
§  74  is  easily  verified. 

Examples  of  well-ordered  series 

77.  The  simplest  examples  of  well-ordered  series  are  those  which 
contain  only  a  finite  number  of  elements;  and  since  two  finite 
series  are  ordinally  similar  when  and  only  when  they  have  the  same 
number  of  elements,  there  will  be  a  distinct  type  of  well-ordered 
series  corresponding  to  every  natural  number  (compare  §  27). 

The  simplest  example  of  a  well-ordered  series  with  an  infinite 
number  of  elements  is  a  series  of  type  co,  that  is,  a  progression  (§  24) . 

78.  Other  examples  of  well-ordered  series,  which  will  serve  also 
to  explain  the  notation  commonly  used,  are  the  following: 

A  progression  of  series  each  of  which  is  itself  of  type  oj  forms  a 
series  of  type  w^ : 

1,  2,  3,  ...  I  1,  2,  3,  ...  I  1,  2,  3,  ...  I 

A  progression  of  series  each  of  which  is  of  type  w^  forms  a  series  of 
type  ca^: 

1,  2,  .  .  I  1,  2,  .  .  i  .  .  II  1,  2,  .  .  1  1,  2,  .  .  1  .  .  II  1,  2,  .  .  I  2,  2,  .  .  I  .  .  II  .  .  .  . 

So  in  general ;  a  progression  of  series  each  of  which  is  of  type  w" 
forms  a  series  of  type  o)"^^,  where  v  is  any  positive  integer. 

Any  type  w"  can  be  represented  by  a  series  of  points  on  a  line  of 
length  a  by  the  following  device,  illustrated  for  the  case  of  type  o)^. 

I 1 1 1— l-i-l 


First,  divide  the  given  line  into  a  denumerable  set  of  intervals,  as 
most  conveniently  by  the  set  of  points  whose  distances  from  the 
right-hand  end  of  the  line  are 

a     a     a      a 

2'    4'    8'    16'  ■  '  " 

*  Jourdain,  Phil.  Mag.,  eer.  6,  vol.  7,  p.  65  (1904). 


§79  WELL-ORDERED  SERIES  67 

the  points  of  division  will  form  a  series  of  type  oj.  Next,  divide 
each  interval  into  a  denumerable  set  of  intervals  in  a  similar  way ; 
all  the  points  of  division  taken  together  will  form  a  series  of  type 
oi^.  Finally,  repeating  the  same  operation  once  again,  we  obtain  a 
series  of  points  of  type  co^. 

79.  A  series  of  the  type  called  w'^  may  now  be  constructed  as 
follows:  Take  a  line  of  length  a,  and  divide  it  into  a  denumerable 
set  of  intervals  as  above;  in  the  first  of  these  intervals  insert  a 
series  of  type  co,  in  the  second  a  series  of  type  co^,  in  the  third  a  series 
of  type  00^,  and  so  on;  the  total  collection  of  points  thus  determined 
forms  a  series  of  type  co"'. 

A  series  of  type  co"  each  of  whose  elements  is  a  series  of  type  co" 
forms  a  series  of  type  (co")^  or  co"'^. 

A  series  of  type  a'^  each  of  whose  elements  is  a  series  of  type 
co"'^  forms  a  series  of  type  co"'^. 

And  so  in  general  a  series  of  type  co"  each  of  whose  elements  is  a 
series  of  type  co""  forms  a  series  of  type  co"(''  +  ^). 

A  series  of  the  type  called  co""  can  now  be  constructed  as  follows : 
Divide  a  given  Hne  into  a  denumerable  set  of  intervals  as  before; 
in  the  first  of  these  intervals  insert  a  series  of  type  co",  in  the  second 
a  series  of  type  co"'^,  in  the  third  a  series  of  type  co"'^,  and  so  on ;  the 
total  collection  of  points  thus  determined  forms  a  series  of  type 
CO"'"  or  CO"". 

A  series  of  type  co"^  each  of  whose  elements  is  a  series  of  type  co"' 
forms  a  series  of  type  (co"')^  or  co""^. 

A  series  of  type  co"'  each  of  whose  elements  is  a  series  of  type 
co"''^  forms  a  series  of  type  co""^. 

And  so  in  general  a  series  of  type  co"'"  may  be  constructed,  and 
hence  a  series  of  the  type  co"'"  or  co"',  by  another  application  of  the 
denumerable  set  of  intervals. 

By  an  extension  of  the  same  methods  we  can  thus  construct 
series  of  each  of  the  types  originally  denoted  by  coi,  co2,  C03,  .  .  ., 
where  ^^  _  ^^^     ^^  =  co"i,     C03  =  co"2,     .  .  .  .* 

*  Cantor,  loc.  cit.  (1897),  p.  242.  It  should  be  noted  that  this  notation  has 
recently  been  abandoned,  the  subscripts  under  the  w's  being  now  used  for 
another  purpose;  see  §  83. 


68  TYPES  OF  SERIAL  ORDER  §  80 

And  so  on  ad  infinitum;  but  none  of  the  well-ordered  series  thus 
constructed  will  contain  more  than  a  denumerable  infinity  of 
elements  (compare  §  38). 

80.  In  order  to  understand  one  further  matter  of  notation,  con- 
sider a  well-ordered  series  of  the  type  represented,  say,  by 

aj3.5  -f  co^7  4-  CO  +  2. 

Here  the  plus  signs  indicate  that  the  series  is  made  up  of  four  parts, 
in  order  from  left  to  right;  the  first  part  consists  of  a  series  of  type 
co^  taken  five  times  in  succession;  the  second  part  consists  of  a 
series  of  type  co^  taken  seven  times  in  succession;  the  third  part  is 
a  single  series  of  type  to;  and  the  last  part  is  a  finite  series  containing 
two  elements.  —  And  so  in  general  the  notation 

where  /x  is  a  positive  integer,  and  the  coefficients  vo,  vi,  V2,  .  .  . ,  v^ 
are  positive  integers  or  zero,  is  to  be  interpreted  in  a  similar  way.* 

It  will  be  noticed  that  in  the  case  of  a  progression,  or  of  any  well- 
ordered  series  of  the  types  described  in  §§  78-79,  the  whole  series  is 
ordinally  similar  to  each  of  its  upper  segments  (§  47) ;  that  is,  if  we 
cut  off  any  lower  segment  from  the  series,  the  type  is  not  altered. 
This  is  not  true  in  the  case  of  the  well-ordered  series  of  the  types 
described  in  the  present  section. 

General  properties  of  well-ordered  series 

81.  The  fundamental  properties  of  well-ordered  series  are  devel- 
oped very  carefully  and  clearly  in  Cantor's  memoir  of  1897;  the 
following  theorems  may  be  mentioned  as  perhaps  the  most  im- 
portant: 

(1)  Every  subclass  in  a  well-ordered  series  is  itself  a  well-ordered 
series. 

(2)  If  each  element  of  a  well-ordered  series  is  replaced  by  a  well- 
ordered  series,  and  the  whole  regarded  as  a  single  series,  the  result 
will  be  still  a  well-ordered  series  (compare  the  examples  in  §§  78- 
79). 

(These  two  theorems  follow  at  once  from  the  definition  in  §  76, 1.) 

*  Cantor,  loc.  cit.  (1897),  p.  229. 


§83  WELL-ORDERED  SERIES  69 

Definition.  The  part  of  a  well-ordered  series  preceding  any 
given  element  a  is  called  a  lower  segment  (Abschnitt)  of  the  series 
(compare  §  47).* 

(3)  A  well-ordered  series  is  never  ordinally  similar  to  any  one 
of  its  lower  segments,  or  to  any  part  of  any  one  of  its  lower 
segments. 

(4)  If  two  well-ordered  series  are  ordinally  similar,  the  ordinal 
correspondence  between  them  can  be  set  up  in  only  one  way  (com- 
pare §§  26,  45,  61,  and  §§  53,  65). 

(5)  Any  subclass  of  a  well-ordered  series  is  ordinally  similar  to 
the  whole  series  or  else  to  some  one  of  its  lower  segments. 

(6)  If  any  two  well-ordered  series,  F  and  G,  are  given,  then  either 
F  is  ordinally  similar  to  G,  or  F  is  ordinally  similar  to  some  definite 
lower  segment  of  G,  or  G  is  ordinally  similar  to  some  definite  lower 
segment  of  F;  and  these  three  relations  are  mutually  exclusive.  In 
the  first  case,  F  and  G  are  of  the  same  type ;  in  the  second  case,  F 
is  said  to  be  less  than  G;  and  in  the  third  case,  G  is  said  to  be  less 
than  F. 

82.  By  virtue  of  this  theorem  6,  the  various  types  of  well-ordered 
series,  when  arranged  "  in  the  order  of  magnitude  "  (as  defined  in  the 
theorem),  form  a  series  (§  74)  with  respect  to  the  relation  "  less  than  "; 
and,  as  Cantor  has  shown,  this  series  is  itself  a  well-ordered  series. 

Moreover,  by  theorem  2,  every  possible  collection  of  types  of 
well-ordered  series,  arranged  in  order  of  magnitude,  will  be  itself 
a  well-ordered  series. 

Classification  of  the  well-ordered  series 

83.  The  classification  of  the  well-ordered  series  is  a  characteristic 
feature  of  Cantor's  theory;  since,  however,  the  method  of  pro- 
cedure, when  pushed  to  its  logical  extreme,  has  led  to  controversy, 

*  Most  writers,  including  Russell,  translate  Abschnitt  by  segment  (without 
qualifying  adjective) ;  but  since  the  word  "  segment  "  is  already  used  in  several 
different  senses  (see,  for  example,  Veblen,  Trans.  Amer.  Math.  Soc.,  vol.  6,  p. 
166,  1905),  it  has  seemed  to  me  safer  to  use  the  longer  term  "  lower  segment," 
about  which  there  can  be  no  ambiguity. 


t   Kr 


alA-XfiX^'U^ 


70  TYPES  OF  SERIAL  ORDER  §83 

the  whole  scheme  is  regarded  with  a  certain  measure  of  suspicion.* 
The  classification  is  as  follows: 

First,  every  well-ordered  series  in  which  the  number  of  elements  is 
finite  is  said  to  belong  to  the  first  class  of  well-ordered  series. 

Now  take  all  the  types  of  series  belonging  to  the  first  class,  and 
arrange  them  in  order  of  magnitude  (§  82) ;  the  result  is  a  well- 
ordered  series  of  a  certain  type,  called  co  (compare  §  24). 

Then  every  well-ordered  series  whose  elements  can  be  put  into  one-to- 
one  correspondence  (§3)  with  the  elements  of  oj  is  said  to  belong  to  the 
SECOND  CLASS.  In  particular,  the  series  of  type  w  are  the  smallest 
series  of  the  second  class. 

Next,  take  all  the  types  of  series  belonging  to  the  second  class, 
and  arrange  them  in  order  of  magnitude;  the  resulting  series  is  a 
well-ordered  series  of  a  certain  type,  called  coi  (or  S2). 

*  On  the  paradoxes  of  Burali-Forti,  Russell,  and  Richard,  and  other  ques- 
tions of  mathematical  logic,  see,  for  example,  C.  BuraU-Forti,  Rend,  del  circ. 
mat.  di  Palermo,  vol.  11  (1897),  pp.  154-164;  E.  Borel,  Lemons  sur  la  theorie  des 
fonctions  (1898),  pp.  119-122,  especially  the  second  edition  (1914),  pp.  102- 
174;  also  a  remark  in  Ldouville's  Journ.  de  Math.,  ser.  5,  vol.  9  (1903),  p.  330; 
D.  Hilbert,  Jahresber.  d.  D.  Math.-Ver.,  vol.  8  (1899),  p.  184;  B.  Russell,  Prin- 
ciples of  Mathematics  (1903),  chapter  10;  E.  W.  Hobson,  Proc.  Land.  Math. 
Soc,  ser.  2,  vol.  3  (1905),  pp.  170-188;  A.  Schonflies  and  A.  Korselt,  Jahresber. 
d.  D.  Math.-Ver.,  vol.  15  (1906),  pp.  19-25  and  215-219;  P.  E.  B.  Jourdain  and 
G.  Peano,  Rivista  di  Matematica,  vol.  8  (1906),  pp.  121-136  and  136-157;  G.  H. 
Hardy,  A.  C.  Dixon,  E.  W.  Hobson,  B.  Russell,  P.  E.  B.  Jomdain,  and  A.  C. 
Dixon,  Proc.  Lond.  Math.  Soc,  ser.  2,  vol.  4  (1906),  pp.  10-17,  18-20,  21-28, 
29-53,  266-283,  and  317-319;  B.  RusseU,  Rev.  de  Metaphys.  et  de  Mor.,  vol. 
14  (1906),  pp.  627-650;  J.  Richard,  Acta  Mathematica,  vol.  30  (1906),  pp.  295- 
296,  and  Ens.  Math.,  vol.  9  (1907),  pp.  94^98;  E.  B.  Wilson,  Bull.  Amer.  Math. 
Soc,  vol.  14  (1908),  pp.  432-443;  A.  Schonflies,  E.  Zermelo,  and  H.  Poin- 
care,  Acta  Mathematica,  vol.  32  (1909),  pp.  177-184,  185-193,  and  195-200; 
A.  Ko>t6  and  B.  Russell,  Rev.  de  Metaphys.  el  de  Mor.,  vol.  20  (1912),  pp.  722- 
724  and  725-726;  H.  Dingier,  Jahresber.  d.  D.  Math.-Ver.,  vol.  22  (1913), 
pp.  307-315;  N.  Wiener,  Messenger  of  Mathematics,  vol.  43  (1913),  pp.  97-105; 
a  curious  paper  by  H.  Glause,  Rend,  del  circ.  mat.  di  Palermo,  vol.  38  (1914), 
pp.  324-329;  and  the  recent  treatises  by  Schonflies,  Konig,  and  Hausdorfi', 
cited  in  a  footnote  to  §  73;  especially  Whitehead  and  Russell,  Principia  Mathe- 
matica, vol.  1  (1910),  pp.  63-68.  On  the  controversy  especially  connected  with 
Zermelo's  "  multiplicative  axiom,"  see  the  references  under  §  84.  On  the 
problem  of  consistency  see  references  under  §  19. 


§84  WELL-ORDERED  SERIES  71 

Then  every  well-ordered  series  whose  elements  can  be  put  into  one- 
to-one  correspondence  with  the  elements  of  coi  is  said  to  belong  to  the 
THIRD  CLASS.  In  particular,  the  series  of  type  oji  are  the  smallest 
series  of  the  third  class. 

And  so  on.  In  general,  every  well-ordered  series  whose  elements  can 
be  put  into  one-to-one  correspondence  with  the  elements  of  co^  (where  v 
is  any  positive  integer)  is  said  to  belong  to  the  (v  +  2)th  class;  and 
the  series  of  type  co^  will  be  the  smallest  series  of  that  class.* 

Moreover,  by  an  extension  of  the  device  already  employed 
several  times,  we  can  define  a  class  of  well-ordered  series  whose 
smallest  type  would  be  denoted  by  co„,  or  even  co„^;  and  so  on,  ml 
infinitum;  so  that  when  we  speak  of  the  nth  class  of  well-ordered 
series,  n  need  not  be  a  positive  integer,  but  may  itself  denote  the 
type  of  any  well  ordered  series. 

84.  In  order  to  justify  this  classification,  it  is  necessary  to  show 
that  the  classes  described  are  really  all  distinct,  so  that  no  well- 
ordered  series  belongs  to  more  than  one  class;  and  further,  that 
well-ordered  series  belonging  to  each  class  actually  exist,  so  that  no 
class  is  "  empty."  Cantor  has  completed  this  investigation  only 
as  far  as  the  first  and  second  classes;  each  of  the  examples  men- 
tioned above  is  a  well-ordered  series  of  the  first  or  second  class 
(since  the  number  of  elements  in  each  case  is  at  most  denumerable, 
in  view  of  §  38) ;  no  similar  example  of  a  series  of  even  the  third 
class  has  yet  been  satisfactorily  constructed. f    Problems  concerning 

*  The  notation  o}„  for  the  smallest  type  of  the  (v  +  2)th  class  was  intro- 
duced by  Russell,  Principles  of  Mathematics,  vol.  1  (1903),  p.  322;  compare 
Jourdain,  Phil.  Mag.,  ser.  6,  vol.  7  (1904),  p.  295.  The  symbols  w  and  i2  were 
first  used  in  this  connection  by  Cantor  in  Math.  Ann.,  vol.  21,  pp.  577,  582 
(1883). 

t  The  question  whether  every  class  can  be  arranged  as  a  well-ordered  series, 
was  first  proposed  by  Cantor  in  1883  {Math.  Ann.,  vol.  21,  p.  550).  The  con- 
troversy centers  about  two  papers  by  E.  Zermelo;  Beweis  doss  jede  Menge 
wohlgeordnet  werden  kann.  Math.  Ann.,  vol.  59  (1904),  pp.  514-516;  Neuer 
Beweis  fur  die  Moglichkeit  einer  Wohlordnung,  Math.  Ann.,  vol.  65  (1907),  pp. 
107-128.  See,  for  example,  J.  Konig,  A.  Schonflies,  F.  Bernstein,  E.  Borel,  and 
P.  E.  B.  Jourdain,  Math.  Ann.,  vol.  60  (1905),  pp.  177,  181,  187,  194,  465; 
J.  Hadamard,  R.  Baire,  H.  Lebesgue,  and  E.  Borel,  Bull,  de  la  Soc.  Math,  de 


72  TYPES  OF  SERIAL  ORDER  §85 

the  existence  of  the  higher  classes,  and  the  question  whether  every 
collection  can  be  arranged  as  a  well-ordered  series,  are  still  being 
actively  debated  (see  §  89). 

85.  The  various  classes  of  well-ordered  series  can  also  be  defined 
by  purely  ordinal  postulates,  as  Veblen  has  shown  how  to  do  in  his 
recent  memoir.* 

Thus,  a  well-ordered  series  of  the  first  class  is  any  well-ordered 
series  which  satisfies  not  only  the  postulates  1-6  of  §  74,  but  also 
the  further  conditions  7i  and  8i,  namely : 

Postulate  7i.  Every  element  except  the  first  has  a  predecessor  (§  17) . 

Postulate  8i.    There  is  a  last  element  (§  17). 

The  type  oj  is  then  defined  by  postulates  1-6  with  7i  and  8'i, 
where  8'i,  is  the  contradictory  of  8i : 

Postulate  8'i.    There  is  no  last  element. 

Next,  a  well-ordered  series  of  the  second  class  is  any  well-ordered 
series,  not  of  the  first  class,  which  satisfies  72  and  82,  namely: 

Postulate  72.  Every  element  except  the  first  either  has  a  predeces- 
sor or  is  the  upper  limit  of  some  subclass  of  type  w  (as  just  defined). 

Postulate  82.  There  is  either  a  last  element,  or  a  subclass  of  type 
CO  which  surpasses  any  given  element  of  the  series,  f 

The  type  coi  (or  fi)  is  then  defined  by  postulates  1-6  with  72  and 
8^2,  where  8'2  is  the  contradictory  of  82. 

Postulate  S'a-  There  is  no  last  element;  and  every  subclass  of 
type  «  has  an  upper  limit  in  the  series. 

France,  vol,  33  (1905),  pp.  261-273;  G.  Peano,  Rivista  di  Matematica,  vol.  8 
(1906),  p.  145;  J.  Konig,  Math.  Ann.,  vol.  61  (1905),  pp.  156-160,  and  vol.  63 
(1906),  pp.  217-221;  H.  Poincar^,  Rev.  de  Metaphys.  et  de  Mor.,  vol.  14  (1906), 
pp.  294-317;  H.  Lebesgue,  Bull,  de  la  Soc.  Math,  de  France,  vol.  35  (1907), 
pp.  202-212;  G.  Vivanti,  Rend,  del  circ.  mat.  di  Palermo,  vol.  25  (1908),  pp.  205- 
208;  G.  Hessenberg,  Crelle's  Journ.  fur  Math.,  vol.  135  (1908),  pp.  81-133, 
318;  E.  Zermelo,  Math.  Ann.,  vol.  65  (1908),  pp.  261-281;  and  the  recent 
treatises  by  Schonflies,  Konig,  and  Hausdorff  cited  in  a  footnote  to  §  73,  espe- 
cially Whitehead  and  Russell,  Princijria  Mathematica,  vol.  3  (1913),  p.  3.  For 
a  third  proof  by  F.  Hartogs  (1915),  see  §  89a. 

*  O.  Veblen,  Trans.  Amer.  Math.  Soc,  vol.  6,  p.  170  (1905). 

t  That  is,  if  X  is  any  element  of  the  given  series,  there  is  an  element  y  in  the 
subclass  for  which  z  "Ky. 


§86  WELL-ORDERED  SERIES  73 

Similarly,  a  well-ordered  series  of  the  third  class  is  any  well- 
ordered  series,  not  of  the  first  or  second  class,  which  satisfies  Ta  and 
83,  namely: 

Postulate  Ts.  Every  element  except  the  first  either  has  a  predeces- 
sor, or  is  the  upper  limit  of  some  subclass  of  type  to,  or  is  the  upper 
limit  of  some  subclass  of  type  oji. 

Postulate  83.  There  is  either  a  last  element,  or  a  subclass  of  type 
w  which  surpasses  any  given  element,  or  a  subclass  of  type  wi  which 
surpasses  any  given  element. 

The  type  W2  is  then  defined  by  postulates  1-6  with  73  and  8'3, 
where,  as  before,  8%  denotes  the  contradictory  of  83 : 

Postulate  8'3.  There  is  no  last  element;  every  subclass  of  type  co 
has  an  upper  limit  in  the  series;  and  every  subclass  of  type  wi  has  an 
upper  limit  in  the  series. 

And  so  on.  The  establishment  of  definite  sets  of  postulates  like 
these  seems  to  me  an  essential  step  toward  the  solution  of  the  diffi- 
cult problems  connected  with  this  subject.  For  example.  Cantor's 
proof  that  a  series  of  type  12  is  non-denumerable  is  simply  a  dem- 
onstration that  no  denumerable  series  can  satisfy  the  eight  postu- 
lates here  numbered  1-6,  72,  and  8'2. 

The  transfinite  ordinal  numbers 

86.  It  is  now  easy  to  explain  what  is  meant  by  the  ordinal  num- 
bers (Ordnungszahlen) ,  in  the  generahzed  sense  in  which  Cantor 
now  uses  that  term:  they  are  simply  the  various  types  of  order  ex- 
hibited by  the  well-ordered  series.*  In  other  words,  according  to  the 
theory  of  Russell,  the  ordinal  number  corresponding  to  any  given 
well-ordered  series  is  the  class  of  all  series  which  are  ordinally  similar 
to  the  given  series;  any  one  of  these  ordinally  similar  series  may  be 
taken  to  represent  the  ordinal  number  of  the  given  series.f 

The  ordinal  numbers  of  the  first  class  (§  83)  are  the  finite  ordinal 
numbers,  with  which  we  have  always  been  familiar;   the  ordinal 

*  Cantor,  Zeitschrift  fiir  Philos.  und  philos.  Kritik,  vol.  91  (1887),  p.  84; 
and  Math.  Ann.,  vol.  49  (1897),  p.  216. 

t  Russell,  Principles  of  Mathematics,  vol.  1  (1903),  p.  312. 


74  TYPES  OF  SERIAL  ORDER  §87 

numbers  of  the  second  or  higher  classes  are  the  transfinite  ordinal 
numbers  created  by  Cantor,  which  constitute,  in  a  certain  true 
sense,  "  eine  Fortsetzung  der  realen  ganzen  Zahlenreihe  iiber  das 
Unendliche  hinaus.^'  * 
The  smallest  of  the  transfinite  ordinals  is  w. 

By  the  sum,  a  -\-  b,  oi  two  ordinal  numbers,  a  and  6,  is  meant 
simply  the  type  of  series  obtained  when  a  series  of  type  a  is  followed 
by  a  series  of  type  b  and  the  whole  regarded  as  a  single  series.f 
Clearly  a  -\-  b  will  not  always  be  the  same  as  6  +  a  (for  example, 
1  +  w  =  oj,  while  CO  +  1  is  a  new  type) ;  but  always  (a  +  6)  +  c  = 
a  +  (6  +  c). 

By  the  product,  ah,  of  an  ordinal  number  a  multiplied  by  an  ordi- 
nal number  b,  is  meant  the  type  of  series  obtained  as  follows :  in  a 
series  of  type  b  replace  each  element  by  a  series  of  type  a,  and  regard 
the  whole  as  a  single  series;  the  result  will  be  a  well-ordered  series 
(by  §  81,  2),  and  the  type  of  this  well-ordered  series  is  what  is 
meant  by  ah. t  Clearly  ah  will  not  always  equal  ba  (for  example, 
2aj  =  CO,  while  00.2  is  a  new  type);  but  always  {ah)c  =  a{bc),  and 
also  a(6  +  c)  =  ah  -\-  ac,  although  not  {b  -\-  c)a  =  ba  +  ca. 

The  definition  of  a*,  where  a  and  b  are  general  ordinal  numbers  is 
too  complicated  to  repeat  in  this  place. §  Enough  has  at  any  rate 
been  said  to  give  at  least  some  notion  of  the  nature  of  the  artificial 
algebra  which  Cantor  has  here  so  ingeniously  constructed. 

The  transfinite  cardinal  numbers 

87.  For  the  sake  of  completeness  I  add  here  a  brief  note  on  the 
meaning  of  some  of  the  terms  in  Cantor's  theory  of  the  (general- 
ized) cardinal  numbers.  ||  This  theory  has  nothing  to  do  with 
series,  or  ordered  classes,  but  is  a  development  of  the  theory  of 
classes  as  such  (§  11);  nevertheless  the  difficulties  met  with  in  this 
theory  are  closely  analogous  to  the  difficulties  we  have  pointed  out 

*  Math.  Ann.,  vol.  21  (1883),  p.  545. 

t  Math.  Ann.,  vol.  21  (1883),  p.  550. 

t  Ib  Cantor's  earlier  definition  of  the  product  ab,  a  was  the  multipUer 
{loc.  cit.,  1883,  p.  551);  the  order  was  changed  in  his  later  articles,  so  that  a 
is  now  the  multiplicand  (see  loc.  cit.,  1887,  p.  96,  and  1897,  pp.  217,  231). 

§  Cantor,  Math.  Ann.,  vol.  49  (1897),  p.  231;  Hausdorff,  loc.  cit.  (1914), 
p.  147. 

II  The  standard  account  of  this  theory  is  in  Cantor's  article  of  1895. 


§88  WELL-ORDERED  SERIES  75 

in  the  theory  of  the  ordinal  numbers  (§  84),  and  it  is  impossible 
to  read  the  literature  of  either  theory  without  some  acquaintance 
with  the  other. 

88.  If  two  classes  can  be  brought  into  one-to-one  correspondence 
(§  3),  they  are  said  to  be  equivalent  (dquivalent) .  For  example,  the 
class  of  rational  numbers  is  equivalent  to  the  class  of  positive 
integers  (compare  §  19,  6) ;  or  the  class  of  points  on  a  line  is  equiv- 
alent to  the  class  of  all  points  in  space  (§71). 

The  cardinal  number  (Mdchtigkeit)  of  a  given  class  A  is  then 
defined  as  the  class  of  all  those  classes  which  are  equivalent  to  A* 
The  finite  cardinal  numbers  are  the  cardinal  numbers  which  belong 
to  finite  classes;  the  transfinite  cardinals  are  those  which  belong  to 
infinite  classes  (§  7). 

According  to  this  definition,  if  two  classes  A  and  B  are  equivalent, 
their  cardinal  numbers  will  clearly  be  identical. 

If  a  class  A  is  equivalent  to  a  part  of  a  class  B,  but  not  to  the 
whole,  then  A  is  said  to  be  less  than  B;  in  this  case  the  cardinal 
number  of  A  will  be  less  than  the  cardinal  number  of  B. 

We  cannot,  however,  affirm  that  all  cardinal  numbers  can  be 
arranged  as  a  series,  in  order  of  magnitude,  for  while  postulates  2 
and  3  (§  74)  clearly  hold  with  regard  to  the  relation  "less  than"  as 
just  defined,  postulate  1,  which  may  be  called  the  principle  of 
comparison  (Vergleichharkeit)  for  classes,  has  never  been  proved. 
In  other  words,  non-equivalent  classes  may  possibly  exist,  neither 
of  which  is  "  less  than  "  the  other;  but  see  §  89a. f 

On  the  other  hand.  Cantor  has  proved  that  when  any  class  is 
given,  a  class  can  be  constructed  which  shall  have  a  greater  cardi- 
nal number  than  the  given  class. | 

*  The  term  Mdchtigkeit  was  first  used  by  Cantor  in  Crelle's  Journ.fiir  Math., 
vol.  84,  p.  242  (1877).  Power,  potency,  multitude,  and  dignity  are  some  of  the 
Enghsh  equivalents.  The  term  Cardinalzahl  was  introduced  in  1887.  Cf. 
Cantor,  loc.  cit.  (1887),  pp.  84  and  118.  The  notion  of  a  cardinal  number  as  a 
doss  is  emphasized  by  Russell;  Principle  of  Mathematics,  vol.  1  (1903),  p.  312. 

t  Compare  E.  Borel,  Leqons  sur  la  theorie  des  fonctions  (1898),  pp.  102-110. 

t  Cantor,  J.  d.  D.  Math.-Ver.,  vol.  1  (1892),  p.  77;  E.  Borel,  loc.  cit.  (1898), 
p.  107;  C.  S.  Peirce,  Monist,  vol.  16  (1906),  pp.  497-502. 


76  TYPES  OF  SERIAL  ORDER  §89 

For  example,  let  C  denote  the  class  of  elements  in  a  linear  con- 
tinuimi,  say  the  class  of  points  on  a  line  one  inch  long  (compare 
§  71) ;  and  let  C  denote  the  class  of  all  possible  "  bi-colored  rods  " 
which  can  be  constructed  by  painting  each  point  of  the  given  line 
either  red  or  blue.  Then  the  class  of  rods,  C,  has  a  higher  cardinal 
number  than  the  class  of  points,  C,  as  may  be  proved  as  follows: 

In  the  first  place,  C  is  equivalent  to  a  'part  of  C ;  for  example,  to 
the  class  of  rods  in  which  one  point  is  painted  red  and  all  the  other 
points  blue.  Secondly,  C  is  not  equivalent  to  the  whole  of  C ;  for,  if 
any  alleged  one-to-one  correspondence  between  the  rods  and  the 
points  were  proposed,  we  could  at  once  define  a  rod  which  would 
not  be  included  in  the  scheme :  namely,  the  rod  in  which  the  color 
of  each  point  x  is  opposite  to  the  color  of  the  point  x  in  the  rod 
which  is  assigned  to  the  point  x  of  the  given  line;  this  rod  would 
differ  from  each  rod  of  the  proposed  scheme  in  the  color  of  at  least 
one  point.     (Cf.  §  40.) 

The  class  C'  has  therefore  a  higher  cardinal  number  than  the  class 
C.  It  is  not  known,  however,  whether  there  may  not  be  other 
classes  whose  cardinal  numbers  lie  between  the  cardinal  numbers  of 
C  and  C. 

89.  Of  special  interest  are  the  cardinal  numbers  of  the  various 
types  of  well-ordered  series;  but  when  we  speak  of  the  cardinal 
number  of  a  series,  it  must  be  understood  that  we  mean  the  cardinal 
number  of  the  class  of  elements  which  occur  in  the  series,  without 
regard  to  their  order. 

The  cardinal  numbers  of  the  finite  well-ordered  series  are  the 
finite  cardinal  numbers,  with  which  we  have  always  been  familiar. 

The  cardinal  number  of  a  series  of  type  co  (§  24)  is  denoted  by 
the  Hebrew  letter  Aleph  with  a  subscript  0  :* 

No. 
This  xo  will  then  be  the  cardinal  number  of  any  well-ordered  series 
of  the  second  class  (§  83),  since  all  the  series  of  the  second  class  are, 
by  definition,  equivalent. 

The  cardinal  number  of  a  series  of  type  coi  (or  fi)  is  denoted  by 
Ni;  this  will  then  be  the  cardmal  number  of  any  weU-ordered 
series  of  the  third  class. 

*  Cantor,  Math.  Ann.,  vol.  46  (1895),  p.  492. 


§89a  WELL-ORDERED  SERIES  77 

And  so  on.  In  general,  the  cardinal  number  of  a  series  of  type 
c>}y  is  denoted  by  a^;  this  will  then  be  the  cardinal  number  of  any 
well-ordered  series  of  the  (v  +  2)th  class. 

//  we  assume  the  series  of  classes  of  ordinal  numbers  (§  84).  we  thus 
obtain  a  series  of  cardinal  numbers 

Ni,  Ki,    .    •    .  ,  Nu>    .   .   .  , 

arranged  in  order  of  increasing  magnitude;  this  series  will  be  a 
well-ordered  series  with  respect  to  the  relation  "  less  than,"  and 
ordinally  similar  to  the  series  of  ordinal  numbers;  but  all  the  diffi- 
culties that  are  involved  in  the  one  series  are  involved  in  the  other. 
In  particular,  it  requires  proof  to  show  that  two  Alephs,  as  K^,  and 
Nh-1,  are  really  non-equivalent,  and  that  no  other  cardinal  number 
lies  between  them.  Cantor  has  shown  merely  that  No  is  the  smallest 
transfinite  cardinal  number,  and  that  Ni  is  the  number  next  greater* 
Again,  the  vexed  question:  can  the  cardinal  number  of  the  linear 
continuum  (§  54)  be  found  among  the  Alephs  f  is  equivalent  to  the 
question :  can  the  class  of  elements  in  the  continuum  be  arranged  in 
the  form  of  a  well-ordered  series  f  (See  §  89a.)  It  is  usually  supposed 
that  the  cardinal  number  of  the  continuum  will  prove  to  be  Xi. 

89a.  In  this  section  we  reproduce,  in  brief  outline,  Hartogs's 
recent  proof  of  Zermelo's  theorem  that  every  class  can  be  arranged 
as  a  well-ordered  series.^ 

Let  there  be  given  any  non-empty  class,  M. 

First,  consider  all  possible  well-ordered  series,  G,  H,  .  .  .  ,  whose 
elements  belong  to  M,  and  let  N  be  the  class  composed  of  these 
series,  together  with  the  null  series,  0. 

Next,  within  this  class  A^,  group  together  all  the  well-ordered 
series  G',  G",  .  .  .  which  are  similar  to  G  into  a  subclass,  g',  group 
together  all  the  well-ordered  series  H',  H",  .  .  .  which  are  similar 
to  H  into  a  subclass,  h;  etc. 

These  subclasses,  g,  h,  .  .  ,  (one  of  which  is  the  null  class)  are 
now  to  form  the  elements  of  a  series,  L,  whose  rule  of  order  is  the 
following:  A  subclass  g  is  said  to  precede  a  subclass  h  (g  <  h),  ii 

*  Math.  Ann.,  vol.  21,  p.  581  (1883). 

t  F.  Hartogs,  Uber  das  Problem  der  Wohlordnung,  Math.  Ann.,  vol.  76 
(1915),  pp.  438-443. 


78  TYPES  OF  SERIAL  ORDER  §  89a 

any  one  of  the  well-ordered  series  belonging  to  g  is  similar  to  a 
lower  segment  of  any  one  of  the  well-ordered  series  H  belonging  to 
h.  (It  is  clear  that  it  makes  no  difference  which  G  is  taken  from 
g,  or  which  H  is  taken  from  h,  etc.,  since  all  the  G's  in  g  are  similar 
to  each  other,  and  all  the  H's  in  h  are  similar  to  each  other,  etc.) 
From  this  definition  it  follows  that  if  any  two  of  the  subclasses,  say 
g  and  h,  are  distinct,  then  either  g  <  hor  else  h  <  g,  and  not  both; 
also  that  if  g,  h,  i  are  three  subclasses  such  that  g  <  h  and  h  <  i, 
thenar  <  i.  In  other  words,  the  subclasses  g,h,  ,  .  ,  form  a  series, 
L,  with  respect  to  the  rule  of  order  stated. 

Moreover,  the  series  L  thus  constructed  is  a  well-ordered  series. 
The  proof  is  as  follows :  Let  g  be  any  element  of  L,  and  let  G  be  any 
one  of  the  well-ordered  series  belonging  to  g.  Then  the  elements  of 
L  which  precede  g  stand  in  a  one-to-one  correspondence  (preserving 
order)  with  the  lower  segments  of  G.  But  the  lower  segments  of  G 
form  a  well-ordered  series;  hence,  no  matter  what  element  g  may 
be  chosen,  the  elements  of  L  preceding  g  form  a  well-ordered  series. 
Froni  this  it  follows  that  the  series  L  itself  must  be  well-ordered. 
For,  if  L  were  not  well-ordered,  it  would  contain  at  least  one  regres- 
sion, r  (§  76),  so  that  if  g  is  any  element  of  r,  then  the  elements  of 
r  preceding  g  would  form  a  series  having  no  first  element;  but  this 
is  impossible,  since  the  elements  of  r  preceding  g  are  part  of  the 
elements  of  L  preceding  g,  and  hence  are  part  of  a  well-ordered 
series,  and  as  such  must  have  a  first  element.  The  whole  series  L 
is  therefore  a  well-ordered  series. 

Further,  each  of  the  well-ordered  series  G,  H,  .  .  .  which  can  be 
formed  out  of  elements  of  M,  is  similar  to  some  lower  segment  of 
L.  In  particular,  the  well-ordered  series  G  is  similar  to  that  lower 
segment  of  L  which  is  determined  by  the  subclass  g  to  which  G 
belongs.  For,  as  we  have  just  noted,  there  is  a  one-to-one  corre- 
spondence (preserving  order)  between  the  subclasses  that  precede 
g  and  the  lower  segments  of  G,  and  there  is  also  a  one-to-one  corre- 
spondence (preserving  order)  between  the  lower  segments  of  G  and 
the  elements  of  G. 

Considering  now  the  elements  of  L,  without  regard  to  their  order, 
we  see  at  once  that  the  elements  of  L  cannot  be  placed  in  one-to-one 
correspondence  with  the  elements  of  M,  nor  with  the  elements  of  any 
part  of  M.  For,  suppose  the  contrary;  then  M,  or  some  part  of  M, 
would  be  capable  of  being  well-ordered,  so  that  we  should  have  a 
well-ordered  series,  formed  out  of  elements  of  M,  and  similar  to  L; 
but  this  is  impossible,  since  we  have  proved  that  every  such  well- 
ordered  series  is  similar  to  some  lower  segment  of  L,  and  no  lower 
segment  of  L  can  be  similar  to  L  itself. 


§  90  WELL-ORDERl^D  SERIES  79 

Finally,  if  we  assume  the  principle  of  comparison  between  classes 
(§  88),  there  is  only  one  alternative  left,  namely:  it  must  he  pos- 
sible to  place  the  elements  of  M  in  one-to-one  correspondence  with 
the  elements  of  a  part  of  L.  But  since  L  is  well-ordered,  every  part 
of  L  is  well-ordered;  hence  we  have  the  theorem  that  whatever 
class  M  may  be,  its  elements  can  always  be  so  arranged  as  to  form 
a  well-ordered  series.* 

90.  We  speak  next  of  the  sums  and  products  of  the  cardinal 
numbers.f 

The  sum  A  -\-  B  oi  two  classes  A  and  B  which  have  no  common 
element  is  the  class  containing  all  the  elements  of  A  and  B  to- 
gether. 

If  a  and  h  are  the  cardinal  numbers  of  two  such  classes  A  and  B, 
the  sum,  a  -\-  h,oi  these  two  cardinals  is  then  defined  as  the  cardinal 
number  oi  A  -\-  B.  Clearly  a  -\-  b  =  b  -\-  a,  and  (a  +  6)  +  c  = 
a  +  (b  +  c). 

The  product,  AB,  of  two  classes  A  and  B  which  have  no  common 
element  is  the  class  of  all  couples  (a,  fi),  where  a  is  any  element  of 
A,  and  /3  any  element  of  B. 

If  a  and  b  are  the  cardinal  numbers  of  two  such  classes,  the  prod- 
uct, ab,  of  these  two  cardinals  is  then  defined  as  the  cardinal  num- 
ber of  AB.  Clearly,  ab  =  ha,  (ab)c  =  a{hc),  and  a{h  +  c)  = 
ah  -\-  ac. 

Finally,  A^  denotes  the  class  of  all  coverings  (Belegungen)  of  B  by 
A,  where  a  "  covering  "  of  B  by  A  is  any  law  according  to  which 
each  element  of  B  determines  uniquely  an  element  of  A  (not  ex- 
cluding the  cases  in  which  various  elements  of  B  may  determine  the 
same  element  of  A).t 

The  ¥''  power  of  a,  a^,  where  a  and  h  are  the  cardinal  numbers  of 
any  two  classes  A  and  B,  is  then  defined  as  the  cardinal  number  of 
A^.     Clearly  a^a"  =  a^",  {a^Y  =  a^',  and  {obY  =  a'=h\ 

In  this  way  Cantor  has  constructed  an  artificial  algebra  of  the 
cardinal  numbers,  analogous  to  the  algebra  of  the  ordinal  numbers, 

*  Hartogs's  paper  shows  that  the  following  three  principles  are  equivalent: 
(1)  the  principle  of  comparison  between  classes;  (2)  the  principle  that  every 
class  can  be  well-ordered;  and  (3)  the  much  discussed  " multiphcative  axiom" 
of  Zermelo.  See  references  under  §  84,  especially  Whitehead  and  RusseU, 
Principia  Mathematica,  vol.  1  (1910),  p.  561. 

t  Zdtschr.f.  Phil.  u.philos.Kritik,  \ol  91  (1887),  pp.  120-121;  Math.  Ann., 
vol.  46  (1895),  p.  485. 

t  Math.  Ann.,  vol.  46  (1895),  p.  487. 


80  TYPES  OF  SERIAL  ORDER  §  91 

but  resembling  much  more  closely  the  familiar  algebra  of  the 
finite  integers. 

Perhaps  the  most  famous  result  obtained  in  this  algebra  is  the 
formula* 

c  =  2^0,  ,^ 

where  c  stands  for  the  cardinal  number  of  the  continuum,  and  2^0 
is  determined  according  to  the  rule  just  stated  for  the  powers  of 
cardinal  numbers.  It  becomes  an  important  question,  therefore, 
to  decide  whether 

2«o  =  Ki 

or  not  (compare  §  89,  end). 

91.  In  conclusion,  it  may  be  well  to  repeat  that  when  we  speak 
of  a  cardinal  number,  we  always  mean  the  cardinal  number  of  some 
given  class;  and  when  we  speak  of  an  ordinal  niunber,  we  always 
mean  the  ordinal  number  of  some  given  well-ordered  series. 

Whether  these  new  concepts  will  find  important  appHcations  in 
practical  problems  is  a  question  for  the  future  to  decide.  (The 
elementary  parts  of  Cantor's  work  have  already  proved  useful, 
indeed  almost  indispensable,  in  the  theory  of  functions  of  a  real 
variable,  t) 

*  Math.  Ann.,  vol.  46  (1895),  p.  488. 

t  See,  for  example,  R.  Baire,  Legons  sur  les  fonctions  discontinues  (1905); 
E.  Borel,  Legons  sur  la  iheorie  des  fonctions,  2nd  edit.  (1914);  E.  W.  Hobson, 
Theory  of  Functions  of  a  Real  Variable  (1907);  J.  Pierpont,  Lectures  on  the 
Theory  of  Functions  of  a  Real  Variable  (1905,  1912);  etc.;  also  the  treatises 
cited  under  §  73. 


INDEX  OF  TECHNICAL  TERMS 


The  principal  bibliographical  footnotes  will  be  found  under  the  introduction,  and 
under  §§  73-74,  and  §§  83-84. 


AlephB,  §  89. 

Between,  §  17. 
Binary  fractions,  §  30. 
Bound  (upper  and  lower),  §  56. 

Cardinal  numbers,  §  88. 

Class,  §  1.  (See  empty,  nvdl,  finite, 
infinite,  denumerable,  simply  and 
multiply  ordered,  well-ordered, 
equivalent.) 

Classes  of  transfinites,  §§  86,  89. 

Closed  (series),  §  62. 

(set  of  points),  §  62a. 

Cluster  point,  §  62a. 

Compact  (series),  §  41. 

(set  of  points),  §  62a. 

Comparison  (of  classes),  §  88. 

Consistency  (of  postulates),  §  19. 

Continued  fractions,  §  71. 

Continuous  (series),  §§  54,  62,  67. 

Continuum,  §§  61,  72. 

problem,  §  89. 

Correspondence  (of  classes),  §  3. 

(of  series),  §  16. 

Covering,  §  90. 

Decimal  fraction,  §§  19  (9),  40,  63  (4). 
Dedekind's  postulate,  §§  21,  54,  75. 
Dense  (series),  §§41,  54,  62. 

(set  of  points),  §  62o. 

Dense-in-itself  (series),  §  62. 
(set  of  points),  §  62a. 


Denumerable  (class),  §  37. 

(series),  §  41. 

Derived  set,  §  62o. 
Digits,  §  30. 

Dimensionality,  §§  67-71. 
Discrete  (series),  §§21,  26. 
Distinct  (elements),  §  2. 

Element  (of  a  class),  §  1.  (See  dis- 
tinct, equal,  first,  last,  rational, 
irrational,  principal,  limit.) 

Empty  (class),  §  1. 

Equal  (elements),  §  2. 

Equivalent  (classes),  §  88. 

Finite  (classes),  §§  7,  27. 

(series),  §  27. 

(numbers),  §§  86,  88. 

First  (element  of  a  series),  §  17. 
Fraction,  §  19.    (See  proper,  decimal, 

binary,  ternary,  continued.) 
Framework  (of  a  series),  §§  59,  67. 
Fimdamental  (segment),  §  46. 
(sequence),  §  62. 

Independence  (of  postulates),  §  20. 

Induction,  §  23. 

Infinite  (classes),  §§  7,  27. 

(numbers),  §§  86,  88. 

Integral  (numbers),  §§  22,  34,  63  (3). 
Irrational  (elements),  §  59. 
(numbers),  §  63  (3). 


81 


82 


INDEX  OF  TECHNICAL  TERMS 


Last  (element  of  a  series),  §  17. 
Less  than,  §§  82,  88. 
Limit  (series),  §§  49,  56,  74. 

(set  of  points),  §  62a. 

Linear  (continuous  series),  §  54. 

Mathematical  induction,  §  23. 
Multiply  ordered  (class),  §  72. 
Multiplicative  axiom,  §  89a. 

Natural  numbers,  §§  19  (1),  30,  36. 

Normal  (series),  §  74. 

Normally  ordered  (class),  §  74. 

NuU  (class),  §  1. 

Numbers,  §  63  (3).  (See  natural,  inte- 
gral, fractional,  rational,  irrational, 
real,  cardinal,  ordinal,  finite,  trans- 
finite.) 

Numeration,  §  30. 

Operations,  §§  11,  53,  65. 

on  natural  numbers,  §§  31,  35. 

on  transfinites,  §§  86,  90. 

Order,  §§  12,  16,  72,  82. 
Ordinal  numbers,  §  86. 
Ordinally  similar  (series),  §  16. 
Origin,  §  26. 

Part  (of  a  class),  §  6. 
Perfect  (series),  §  62. 

(set  of  points),  §  62a. 

Point  sets,  §  62a. 

Postulates,  §§  12,  21,  41,  54,  74,  85. 

consistency  of,  §  19. 

independence  of,  §  20. 

Powers  (of  numbers) .    See  operations. 

(cardinals),  §  88. 

Predecessor,  §  17. 

Principal  (element  of  a  series),  §  62. 

Products.     See  operations. 

Progression,  §§  24,  85. 

Proper  fraction,  §§  19  (5),  42. 


Rational  (elements),  §  59. 

(numbers),  §§  51,  63  (3). 

Real  (numbers),  §§63  (3). 
Regression,  §  25. 
Relation,  §§  11,  12,  13. 

Section  (of  a  continuous  series),  §  68. 

Segment,  §  47. 

(fundamental),  §  46. 

(upper  and  lower),  §  47. 

(well-ordered  series),  §  81. 

Self -representative,  §  28. 

Sequence,  §  62. 

Series,  §  12.  (See  discrete,  dense, 
denumerable,  continuous,  linear, 
finite,  closed,  dense-in-itself,  per- 
fect, well-ordered,  similar.) 

Sets  of  points,  §  62a. 

Similar  (series),  §  16. 

Simply  ordered  (class),  §  12. 

Skeleton  (of  a  series),  §§  59,  67. 

Subclass,  §  6. 

Successor,  §  17. 

Sums.     See  operations. 

System,  §  11. 

Ternary  fractions,  §  52  (3). 
Transfinite  numbers,  §§  86,  88. 
Types  of  order,  §  16. 
Type  CO,  §§  24,  85. 

*co,  §25. 

*«-H«,  §26. 

V,  §  44. 

0,    §§61,62. 

0^  §  69. 

co^  «-,  §§  78,  79. 

4o„  §§  79,  83,  85. 

n,  §§  83,  85. 

Well-ordered  (series),  §§  74,  76. 


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